Properties

Label 4-10e2-1.1-c21e2-0-1
Degree 44
Conductor 100100
Sign 11
Analytic cond. 781.075781.075
Root an. cond. 5.286565.28656
Motivic weight 2121
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.04e3·2-s + 3.09e4·3-s + 3.14e6·4-s − 1.95e7·5-s + 6.34e7·6-s − 4.39e8·7-s + 4.29e9·8-s − 3.21e9·9-s − 4.00e10·10-s + 1.05e11·11-s + 9.74e10·12-s + 3.08e11·13-s − 9.01e11·14-s − 6.04e11·15-s + 5.49e12·16-s + 1.83e13·17-s − 6.58e12·18-s + 2.38e13·19-s − 6.14e13·20-s − 1.36e13·21-s + 2.15e14·22-s − 8.25e13·23-s + 1.33e14·24-s + 2.86e14·25-s + 6.31e14·26-s + 9.51e13·27-s − 1.38e15·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.302·3-s + 3/2·4-s − 0.894·5-s + 0.428·6-s − 0.588·7-s + 1.41·8-s − 0.307·9-s − 1.26·10-s + 1.22·11-s + 0.454·12-s + 0.620·13-s − 0.832·14-s − 0.270·15-s + 5/4·16-s + 2.21·17-s − 0.434·18-s + 0.891·19-s − 1.34·20-s − 0.178·21-s + 1.72·22-s − 0.415·23-s + 0.428·24-s + 3/5·25-s + 0.877·26-s + 0.0889·27-s − 0.883·28-s + ⋯

Functional equation

Λ(s)=(100s/2ΓC(s)2L(s)=(Λ(22s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}
Λ(s)=(100s/2ΓC(s+21/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 781.075781.075
Root analytic conductor: 5.286565.28656
Motivic weight: 2121
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 100, ( :21/2,21/2), 1)(4,\ 100,\ (\ :21/2, 21/2),\ 1)

Particular Values

L(11)L(11) \approx 8.4829909078.482990907
L(12)L(\frac12) \approx 8.4829909078.482990907
L(232)L(\frac{23}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1p10T)2 ( 1 - p^{10} T )^{2}
5C1C_1 (1+p10T)2 ( 1 + p^{10} T )^{2}
good3D4D_{4} 110324pT+17175214p5T210324p22T3+p42T4 1 - 10324 p T + 17175214 p^{5} T^{2} - 10324 p^{22} T^{3} + p^{42} T^{4}
7D4D_{4} 1+439959356T+166460827342795614pT2+439959356p21T3+p42T4 1 + 439959356 T + 166460827342795614 p T^{2} + 439959356 p^{21} T^{3} + p^{42} T^{4}
11D4D_{4} 1105191777184T+ 1 - 105191777184 T + 74 ⁣ ⁣2674\!\cdots\!26pT2105191777184p21T3+p42T4 p T^{2} - 105191777184 p^{21} T^{3} + p^{42} T^{4}
13D4D_{4} 1308456648932T+ 1 - 308456648932 T + 31 ⁣ ⁣1431\!\cdots\!14pT2308456648932p21T3+p42T4 p T^{2} - 308456648932 p^{21} T^{3} + p^{42} T^{4}
17D4D_{4} 11081999770612pT+ 1 - 1081999770612 p T + 72 ⁣ ⁣4272\!\cdots\!42p2T21081999770612p22T3+p42T4 p^{2} T^{2} - 1081999770612 p^{22} T^{3} + p^{42} T^{4}
19D4D_{4} 11253748248200pT+ 1 - 1253748248200 p T + 28 ⁣ ⁣5828\!\cdots\!58p2T21253748248200p22T3+p42T4 p^{2} T^{2} - 1253748248200 p^{22} T^{3} + p^{42} T^{4}
23D4D_{4} 1+82586042978868T+ 1 + 82586042978868 T + 77 ⁣ ⁣0277\!\cdots\!02T2+82586042978868p21T3+p42T4 T^{2} + 82586042978868 p^{21} T^{3} + p^{42} T^{4}
29D4D_{4} 1+167038888446420T+ 1 + 167038888446420 T + 27 ⁣ ⁣5827\!\cdots\!58T2+167038888446420p21T3+p42T4 T^{2} + 167038888446420 p^{21} T^{3} + p^{42} T^{4}
31D4D_{4} 15373084998145784T+ 1 - 5373084998145784 T + 46 ⁣ ⁣2646\!\cdots\!26T25373084998145784p21T3+p42T4 T^{2} - 5373084998145784 p^{21} T^{3} + p^{42} T^{4}
37D4D_{4} 183725084127912164T+ 1 - 83725084127912164 T + 32 ⁣ ⁣9832\!\cdots\!98T283725084127912164p21T3+p42T4 T^{2} - 83725084127912164 p^{21} T^{3} + p^{42} T^{4}
41D4D_{4} 1+63222375005514036T+ 1 + 63222375005514036 T + 15 ⁣ ⁣0615\!\cdots\!06T2+63222375005514036p21T3+p42T4 T^{2} + 63222375005514036 p^{21} T^{3} + p^{42} T^{4}
43D4D_{4} 1+113835841911164948T+ 1 + 113835841911164948 T + 12 ⁣ ⁣6212\!\cdots\!62T2+113835841911164948p21T3+p42T4 T^{2} + 113835841911164948 p^{21} T^{3} + p^{42} T^{4}
47D4D_{4} 113253549001226164T+ 1 - 13253549001226164 T + 24 ⁣ ⁣1824\!\cdots\!18T213253549001226164p21T3+p42T4 T^{2} - 13253549001226164 p^{21} T^{3} + p^{42} T^{4}
53D4D_{4} 1+1445751023743904748T+ 1 + 1445751023743904748 T + 67 ⁣ ⁣8267\!\cdots\!82T2+1445751023743904748p21T3+p42T4 T^{2} + 1445751023743904748 p^{21} T^{3} + p^{42} T^{4}
59D4D_{4} 1+817060118931432840T+ 1 + 817060118931432840 T + 30 ⁣ ⁣1830\!\cdots\!18T2+817060118931432840p21T3+p42T4 T^{2} + 817060118931432840 p^{21} T^{3} + p^{42} T^{4}
61D4D_{4} 1+4580997169825849436T+ 1 + 4580997169825849436 T + 32 ⁣ ⁣4632\!\cdots\!46T2+4580997169825849436p21T3+p42T4 T^{2} + 4580997169825849436 p^{21} T^{3} + p^{42} T^{4}
67D4D_{4} 1+28808031668773210556T+ 1 + 28808031668773210556 T + 64 ⁣ ⁣1864\!\cdots\!18T2+28808031668773210556p21T3+p42T4 T^{2} + 28808031668773210556 p^{21} T^{3} + p^{42} T^{4}
71D4D_{4} 151140957732016114984T+ 1 - 51140957732016114984 T + 21 ⁣ ⁣0621\!\cdots\!06T251140957732016114984p21T3+p42T4 T^{2} - 51140957732016114984 p^{21} T^{3} + p^{42} T^{4}
73D4D_{4} 114792322512082321412T+ 1 - 14792322512082321412 T + 38 ⁣ ⁣8238\!\cdots\!82T214792322512082321412p21T3+p42T4 T^{2} - 14792322512082321412 p^{21} T^{3} + p^{42} T^{4}
79D4D_{4} 125077766877525032720T+ 1 - 25077766877525032720 T + 11 ⁣ ⁣5811\!\cdots\!58T225077766877525032720p21T3+p42T4 T^{2} - 25077766877525032720 p^{21} T^{3} + p^{42} T^{4}
83D4D_{4} 1+ 1 + 12 ⁣ ⁣6812\!\cdots\!68T+ T + 38 ⁣ ⁣2238\!\cdots\!22T2+ T^{2} + 12 ⁣ ⁣6812\!\cdots\!68p21T3+p42T4 p^{21} T^{3} + p^{42} T^{4}
89D4D_{4} 1 1 - 10 ⁣ ⁣8010\!\cdots\!80T T - 33 ⁣ ⁣2233\!\cdots\!22T2 T^{2} - 10 ⁣ ⁣8010\!\cdots\!80p21T3+p42T4 p^{21} T^{3} + p^{42} T^{4}
97D4D_{4} 1+ 1 + 26 ⁣ ⁣9626\!\cdots\!96T+ T + 45 ⁣ ⁣9845\!\cdots\!98T2+ T^{2} + 26 ⁣ ⁣9626\!\cdots\!96p21T3+p42T4 p^{21} T^{3} + p^{42} T^{4}
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   L(s)=p j=14(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−15.80904391050943419199518934072, −15.13882234462509989118536923004, −14.30191180182851090555111216750, −14.16960218827182429796380241887, −13.11501084736210432123292800820, −12.44505953192757271287888041570, −11.61423277013174624547472955210, −11.53921826035046864923926285742, −10.16505969652227915066826270464, −9.426914530971558626016586784040, −8.100065386988730858243015776923, −7.63250874819259180382847280239, −6.45860842729809545368635697469, −6.01437335541878318939133969791, −4.89552429151551962596653638970, −4.05255449788773471739179722733, −3.20267462228395308125262242873, −3.10198697336722944316667572319, −1.48514684380487484315958066096, −0.794842427459680101765561788939, 0.794842427459680101765561788939, 1.48514684380487484315958066096, 3.10198697336722944316667572319, 3.20267462228395308125262242873, 4.05255449788773471739179722733, 4.89552429151551962596653638970, 6.01437335541878318939133969791, 6.45860842729809545368635697469, 7.63250874819259180382847280239, 8.100065386988730858243015776923, 9.426914530971558626016586784040, 10.16505969652227915066826270464, 11.53921826035046864923926285742, 11.61423277013174624547472955210, 12.44505953192757271287888041570, 13.11501084736210432123292800820, 14.16960218827182429796380241887, 14.30191180182851090555111216750, 15.13882234462509989118536923004, 15.80904391050943419199518934072

Graph of the ZZ-function along the critical line