Properties

Label 4-10e2-1.1-c23e2-0-0
Degree 44
Conductor 100100
Sign 11
Analytic cond. 1123.611123.61
Root an. cond. 5.789685.78968
Motivic weight 2323
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.09e3·2-s + 6.86e5·3-s + 1.25e7·4-s − 9.76e7·5-s − 2.81e9·6-s − 3.52e9·7-s − 3.43e10·8-s + 2.25e11·9-s + 4.00e11·10-s + 9.36e11·11-s + 8.63e12·12-s + 9.96e12·13-s + 1.44e13·14-s − 6.70e13·15-s + 8.79e13·16-s − 1.60e12·17-s − 9.25e14·18-s − 2.56e14·19-s − 1.22e15·20-s − 2.42e15·21-s − 3.83e15·22-s + 3.59e15·23-s − 2.35e16·24-s + 7.15e15·25-s − 4.08e16·26-s + 5.13e16·27-s − 4.44e16·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.23·3-s + 3/2·4-s − 0.894·5-s − 3.16·6-s − 0.674·7-s − 1.41·8-s + 2.40·9-s + 1.26·10-s + 0.989·11-s + 3.35·12-s + 1.54·13-s + 0.954·14-s − 2.00·15-s + 5/4·16-s − 0.0113·17-s − 3.39·18-s − 0.505·19-s − 1.34·20-s − 1.50·21-s − 1.39·22-s + 0.786·23-s − 3.16·24-s + 3/5·25-s − 2.18·26-s + 1.77·27-s − 1.01·28-s + ⋯

Functional equation

Λ(s)=(100s/2ΓC(s)2L(s)=(Λ(24s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}
Λ(s)=(100s/2ΓC(s+23/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 1123.611123.61
Root analytic conductor: 5.789685.78968
Motivic weight: 2323
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 100, ( :23/2,23/2), 1)(4,\ 100,\ (\ :23/2, 23/2),\ 1)

Particular Values

L(12)L(12) \approx 4.0418661364.041866136
L(12)L(\frac12) \approx 4.0418661364.041866136
L(252)L(\frac{25}{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 (1+p11T)2 ( 1 + p^{11} T )^{2}
5C1C_1 (1+p11T)2 ( 1 + p^{11} T )^{2}
good3D4D_{4} 176276p2T+112148014p7T276276p25T3+p46T4 1 - 76276 p^{2} T + 112148014 p^{7} T^{2} - 76276 p^{25} T^{3} + p^{46} T^{4}
7D4D_{4} 1+3529595108T+210532244602639698p2T2+3529595108p23T3+p46T4 1 + 3529595108 T + 210532244602639698 p^{2} T^{2} + 3529595108 p^{23} T^{3} + p^{46} T^{4}
11D4D_{4} 185141569984pT+ 1 - 85141569984 p T + 11 ⁣ ⁣8611\!\cdots\!86p2T285141569984p24T3+p46T4 p^{2} T^{2} - 85141569984 p^{24} T^{3} + p^{46} T^{4}
13D4D_{4} 19961846393684T+ 1 - 9961846393684 T + 89 ⁣ ⁣5889\!\cdots\!58T29961846393684p23T3+p46T4 T^{2} - 9961846393684 p^{23} T^{3} + p^{46} T^{4}
17D4D_{4} 1+94688808204pT+ 1 + 94688808204 p T + 81 ⁣ ⁣3881\!\cdots\!38p2T2+94688808204p24T3+p46T4 p^{2} T^{2} + 94688808204 p^{24} T^{3} + p^{46} T^{4}
19D4D_{4} 1+256914240498200T+ 1 + 256914240498200 T + 19 ⁣ ⁣2219\!\cdots\!22pT2+256914240498200p23T3+p46T4 p T^{2} + 256914240498200 p^{23} T^{3} + p^{46} T^{4}
23D4D_{4} 13594102326172564T+ 1 - 3594102326172564 T + 40 ⁣ ⁣5840\!\cdots\!58T23594102326172564p23T3+p46T4 T^{2} - 3594102326172564 p^{23} T^{3} + p^{46} T^{4}
29D4D_{4} 1195470005256002620T+ 1 - 195470005256002620 T + 17 ⁣ ⁣7817\!\cdots\!78T2195470005256002620p23T3+p46T4 T^{2} - 195470005256002620 p^{23} T^{3} + p^{46} T^{4}
31D4D_{4} 1168811393572212104T+ 1 - 168811393572212104 T + 47 ⁣ ⁣8647\!\cdots\!86T2168811393572212104p23T3+p46T4 T^{2} - 168811393572212104 p^{23} T^{3} + p^{46} T^{4}
37D4D_{4} 1914879862377868532T+ 1 - 914879862377868532 T + 91 ⁣ ⁣6291\!\cdots\!62T2914879862377868532p23T3+p46T4 T^{2} - 914879862377868532 p^{23} T^{3} + p^{46} T^{4}
41D4D_{4} 11944225537689591724T+ 1 - 1944225537689591724 T + 19 ⁣ ⁣8619\!\cdots\!86T21944225537689591724p23T3+p46T4 T^{2} - 1944225537689591724 p^{23} T^{3} + p^{46} T^{4}
43D4D_{4} 12162030450882223364T+ 1 - 2162030450882223364 T + 65 ⁣ ⁣3865\!\cdots\!38T22162030450882223364p23T3+p46T4 T^{2} - 2162030450882223364 p^{23} T^{3} + p^{46} T^{4}
47D4D_{4} 120899427956391479692T+ 1 - 20899427956391479692 T + 33 ⁣ ⁣6233\!\cdots\!62T220899427956391479692p23T3+p46T4 T^{2} - 20899427956391479692 p^{23} T^{3} + p^{46} T^{4}
53D4D_{4} 172675644730523142244T+ 1 - 72675644730523142244 T + 73 ⁣ ⁣3873\!\cdots\!38T272675644730523142244p23T3+p46T4 T^{2} - 72675644730523142244 p^{23} T^{3} + p^{46} T^{4}
59D4D_{4} 1+ 1 + 26 ⁣ ⁣6026\!\cdots\!60T+ T + 12 ⁣ ⁣5812\!\cdots\!58T2+ T^{2} + 26 ⁣ ⁣6026\!\cdots\!60p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
61D4D_{4} 1+ 1 + 34 ⁣ ⁣9634\!\cdots\!96T+ T + 25 ⁣ ⁣6625\!\cdots\!66T2+ T^{2} + 34 ⁣ ⁣9634\!\cdots\!96p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
67D4D_{4} 1 1 - 17 ⁣ ⁣5217\!\cdots\!52T+ T + 21 ⁣ ⁣0221\!\cdots\!02T2 T^{2} - 17 ⁣ ⁣5217\!\cdots\!52p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
71D4D_{4} 1 1 - 57 ⁣ ⁣6457\!\cdots\!64T+ T + 15 ⁣ ⁣4615\!\cdots\!46T2 T^{2} - 57 ⁣ ⁣6457\!\cdots\!64p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
73D4D_{4} 1+ 1 + 10 ⁣ ⁣7610\!\cdots\!76T+ T + 12 ⁣ ⁣7812\!\cdots\!78T2+ T^{2} + 10 ⁣ ⁣7610\!\cdots\!76p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
79D4D_{4} 1+ 1 + 51 ⁣ ⁣2051\!\cdots\!20T+ T + 56 ⁣ ⁣7856\!\cdots\!78T2+ T^{2} + 51 ⁣ ⁣2051\!\cdots\!20p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
83D4D_{4} 1 1 - 87 ⁣ ⁣4487\!\cdots\!44T+ T + 28 ⁣ ⁣5828\!\cdots\!58T2 T^{2} - 87 ⁣ ⁣4487\!\cdots\!44p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
89D4D_{4} 1 1 - 24 ⁣ ⁣2024\!\cdots\!20T+ T + 14 ⁣ ⁣3814\!\cdots\!38T2 T^{2} - 24 ⁣ ⁣2024\!\cdots\!20p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
97D4D_{4} 1 1 - 28 ⁣ ⁣1228\!\cdots\!12T+ T + 66 ⁣ ⁣8266\!\cdots\!82T2 T^{2} - 28 ⁣ ⁣1228\!\cdots\!12p23T3+p46T4 p^{23} T^{3} + p^{46} 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

−15.50997745658585319492659493277, −15.19117359675760961223231492886, −14.13413588294952209331189303042, −13.83123498308352105226391530833, −12.69464495104874652409336596793, −11.93366291452725969285456019802, −10.94404979533563908472560842663, −10.21339686114719998896677824859, −9.152373569808496861084334929293, −8.993758261191315894266003781862, −8.115597605634270287552641704517, −8.101830958496358041931908930425, −6.77114373542903646842451922125, −6.41371313771799990709877614830, −4.27934146385728398294270129474, −3.56112422779799500224801165446, −2.88858060961328901916219702569, −2.40954745454556075414213239421, −1.05751238210411616523404130839, −0.863674310779455764603152532422, 0.863674310779455764603152532422, 1.05751238210411616523404130839, 2.40954745454556075414213239421, 2.88858060961328901916219702569, 3.56112422779799500224801165446, 4.27934146385728398294270129474, 6.41371313771799990709877614830, 6.77114373542903646842451922125, 8.101830958496358041931908930425, 8.115597605634270287552641704517, 8.993758261191315894266003781862, 9.152373569808496861084334929293, 10.21339686114719998896677824859, 10.94404979533563908472560842663, 11.93366291452725969285456019802, 12.69464495104874652409336596793, 13.83123498308352105226391530833, 14.13413588294952209331189303042, 15.19117359675760961223231492886, 15.50997745658585319492659493277

Graph of the ZZ-function along the critical line