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Properties

Label 4-10e2-1.1-c21e2-0-1
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $781.075$
Root an. cond. $5.28656$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

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

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

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(781.075\)
Root analytic conductor: \(5.28656\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 100,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(8.482990907\)
\(L(\frac12)\) \(\approx\) \(8.482990907\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - p^{10} T )^{2} \)
5$C_1$ \( ( 1 + p^{10} T )^{2} \)
good3$D_{4}$ \( 1 - 10324 p T + 17175214 p^{5} T^{2} - 10324 p^{22} T^{3} + p^{42} T^{4} \)
7$D_{4}$ \( 1 + 439959356 T + 166460827342795614 p T^{2} + 439959356 p^{21} T^{3} + p^{42} T^{4} \)
11$D_{4}$ \( 1 - 105191777184 T + \)\(74\!\cdots\!26\)\( p T^{2} - 105191777184 p^{21} T^{3} + p^{42} T^{4} \)
13$D_{4}$ \( 1 - 308456648932 T + \)\(31\!\cdots\!14\)\( p T^{2} - 308456648932 p^{21} T^{3} + p^{42} T^{4} \)
17$D_{4}$ \( 1 - 1081999770612 p T + \)\(72\!\cdots\!42\)\( p^{2} T^{2} - 1081999770612 p^{22} T^{3} + p^{42} T^{4} \)
19$D_{4}$ \( 1 - 1253748248200 p T + \)\(28\!\cdots\!58\)\( p^{2} T^{2} - 1253748248200 p^{22} T^{3} + p^{42} T^{4} \)
23$D_{4}$ \( 1 + 82586042978868 T + \)\(77\!\cdots\!02\)\( T^{2} + 82586042978868 p^{21} T^{3} + p^{42} T^{4} \)
29$D_{4}$ \( 1 + 167038888446420 T + \)\(27\!\cdots\!58\)\( T^{2} + 167038888446420 p^{21} T^{3} + p^{42} T^{4} \)
31$D_{4}$ \( 1 - 5373084998145784 T + \)\(46\!\cdots\!26\)\( T^{2} - 5373084998145784 p^{21} T^{3} + p^{42} T^{4} \)
37$D_{4}$ \( 1 - 83725084127912164 T + \)\(32\!\cdots\!98\)\( T^{2} - 83725084127912164 p^{21} T^{3} + p^{42} T^{4} \)
41$D_{4}$ \( 1 + 63222375005514036 T + \)\(15\!\cdots\!06\)\( T^{2} + 63222375005514036 p^{21} T^{3} + p^{42} T^{4} \)
43$D_{4}$ \( 1 + 113835841911164948 T + \)\(12\!\cdots\!62\)\( T^{2} + 113835841911164948 p^{21} T^{3} + p^{42} T^{4} \)
47$D_{4}$ \( 1 - 13253549001226164 T + \)\(24\!\cdots\!18\)\( T^{2} - 13253549001226164 p^{21} T^{3} + p^{42} T^{4} \)
53$D_{4}$ \( 1 + 1445751023743904748 T + \)\(67\!\cdots\!82\)\( T^{2} + 1445751023743904748 p^{21} T^{3} + p^{42} T^{4} \)
59$D_{4}$ \( 1 + 817060118931432840 T + \)\(30\!\cdots\!18\)\( T^{2} + 817060118931432840 p^{21} T^{3} + p^{42} T^{4} \)
61$D_{4}$ \( 1 + 4580997169825849436 T + \)\(32\!\cdots\!46\)\( T^{2} + 4580997169825849436 p^{21} T^{3} + p^{42} T^{4} \)
67$D_{4}$ \( 1 + 28808031668773210556 T + \)\(64\!\cdots\!18\)\( T^{2} + 28808031668773210556 p^{21} T^{3} + p^{42} T^{4} \)
71$D_{4}$ \( 1 - 51140957732016114984 T + \)\(21\!\cdots\!06\)\( T^{2} - 51140957732016114984 p^{21} T^{3} + p^{42} T^{4} \)
73$D_{4}$ \( 1 - 14792322512082321412 T + \)\(38\!\cdots\!82\)\( T^{2} - 14792322512082321412 p^{21} T^{3} + p^{42} T^{4} \)
79$D_{4}$ \( 1 - 25077766877525032720 T + \)\(11\!\cdots\!58\)\( T^{2} - 25077766877525032720 p^{21} T^{3} + p^{42} T^{4} \)
83$D_{4}$ \( 1 + \)\(12\!\cdots\!68\)\( T + \)\(38\!\cdots\!22\)\( T^{2} + \)\(12\!\cdots\!68\)\( p^{21} T^{3} + p^{42} T^{4} \)
89$D_{4}$ \( 1 - \)\(10\!\cdots\!80\)\( T - \)\(33\!\cdots\!22\)\( T^{2} - \)\(10\!\cdots\!80\)\( p^{21} T^{3} + p^{42} T^{4} \)
97$D_{4}$ \( 1 + \)\(26\!\cdots\!96\)\( T + \)\(45\!\cdots\!98\)\( T^{2} + \)\(26\!\cdots\!96\)\( p^{21} T^{3} + p^{42} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line