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Properties

Label 4-20e2-1.1-c37e2-0-0
Degree $4$
Conductor $400$
Sign $1$
Analytic cond. $30077.2$
Root an. cond. $13.1692$
Motivic weight $37$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.24e5·2-s + 1.37e11·4-s + 2.10e12·5-s + 1.10e18·10-s + 9.77e20·13-s − 1.88e22·16-s − 1.49e23·17-s + 2.89e23·20-s − 6.83e25·25-s + 5.12e26·26-s − 9.90e27·32-s − 7.83e28·34-s − 1.67e29·37-s + 4.57e29·41-s − 3.58e31·50-s + 1.34e32·52-s − 2.23e32·53-s + 3.59e33·61-s − 2.59e33·64-s + 2.05e33·65-s − 2.05e34·68-s − 8.35e34·73-s − 8.76e34·74-s − 3.97e34·80-s − 2.02e35·81-s + 2.39e35·82-s − 3.14e35·85-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.246·5-s + 0.349·10-s + 2.41·13-s − 16-s − 2.57·17-s + 0.246·20-s − 0.939·25-s + 3.40·26-s − 1.41·32-s − 3.64·34-s − 1.62·37-s + 0.666·41-s − 1.32·50-s + 2.41·52-s − 2.81·53-s + 3.36·61-s − 64-s + 0.595·65-s − 2.57·68-s − 2.81·73-s − 2.30·74-s − 0.246·80-s − 81-s + 0.942·82-s − 0.636·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+37/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(30077.2\)
Root analytic conductor: \(13.1692\)
Motivic weight: \(37\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 400,\ (\ :37/2, 37/2),\ 1)\)

Particular Values

\(L(19)\) \(\approx\) \(0.2915044547\)
\(L(\frac12)\) \(\approx\) \(0.2915044547\)
\(L(\frac{39}{2})\) 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$C_2$ \( 1 - p^{19} T + p^{37} T^{2} \)
5$C_2$ \( 1 - 2106585175756 T + p^{37} T^{2} \)
good3$C_2^2$ \( 1 + p^{74} T^{4} \)
7$C_2^2$ \( 1 + p^{74} T^{4} \)
11$C_2$ \( ( 1 - p^{37} T^{2} )^{2} \)
13$C_2$ \( ( 1 - \)\(78\!\cdots\!94\)\( T + p^{37} T^{2} )( 1 - \)\(18\!\cdots\!64\)\( T + p^{37} T^{2} ) \)
17$C_2$ \( ( 1 + \)\(40\!\cdots\!62\)\( T + p^{37} T^{2} )( 1 + \)\(10\!\cdots\!92\)\( T + p^{37} T^{2} ) \)
19$C_2$ \( ( 1 + p^{37} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{74} T^{4} \)
29$C_2$ \( ( 1 - \)\(13\!\cdots\!70\)\( T + p^{37} T^{2} )( 1 + \)\(13\!\cdots\!70\)\( T + p^{37} T^{2} ) \)
31$C_2$ \( ( 1 - p^{37} T^{2} )^{2} \)
37$C_2$ \( ( 1 - \)\(35\!\cdots\!38\)\( T + p^{37} T^{2} )( 1 + \)\(20\!\cdots\!32\)\( T + p^{37} T^{2} ) \)
41$C_2$ \( ( 1 - \)\(22\!\cdots\!32\)\( T + p^{37} T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{74} T^{4} \)
47$C_2^2$ \( 1 + p^{74} T^{4} \)
53$C_2$ \( ( 1 + \)\(10\!\cdots\!66\)\( T + p^{37} T^{2} )( 1 + \)\(12\!\cdots\!36\)\( T + p^{37} T^{2} ) \)
59$C_2$ \( ( 1 + p^{37} T^{2} )^{2} \)
61$C_2$ \( ( 1 - \)\(17\!\cdots\!72\)\( T + p^{37} T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{74} T^{4} \)
71$C_2$ \( ( 1 - p^{37} T^{2} )^{2} \)
73$C_2$ \( ( 1 + \)\(38\!\cdots\!06\)\( T + p^{37} T^{2} )( 1 + \)\(44\!\cdots\!76\)\( T + p^{37} T^{2} ) \)
79$C_2$ \( ( 1 + p^{37} T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{74} T^{4} \)
89$C_2$ \( ( 1 - \)\(22\!\cdots\!70\)\( T + p^{37} T^{2} )( 1 + \)\(22\!\cdots\!70\)\( T + p^{37} T^{2} ) \)
97$C_2$ \( ( 1 - \)\(26\!\cdots\!08\)\( T + p^{37} T^{2} )( 1 + \)\(11\!\cdots\!22\)\( T + p^{37} T^{2} ) \)
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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

−11.52443396620969954296692261123, −11.12157962911757596684358503736, −10.94942971943725818731871560682, −9.942438967164571860061575858903, −9.182042293678520906078410111145, −8.522861296168303217599352412099, −8.447837726563350907614761235554, −7.12700243751507901763289643588, −6.72915598277194228048282953125, −5.98860983077545729702679217475, −5.97770604441599695611690690969, −5.05659359186508086286262864110, −4.47064007375122397496663389975, −3.93938232444722287208733073178, −3.61697385384425572294432210274, −2.88641918595823578633739978564, −2.24604759583599668781578290432, −1.70929856412314773300904297277, −1.17336601448765344787987143207, −0.07391704103168566065757599898, 0.07391704103168566065757599898, 1.17336601448765344787987143207, 1.70929856412314773300904297277, 2.24604759583599668781578290432, 2.88641918595823578633739978564, 3.61697385384425572294432210274, 3.93938232444722287208733073178, 4.47064007375122397496663389975, 5.05659359186508086286262864110, 5.97770604441599695611690690969, 5.98860983077545729702679217475, 6.72915598277194228048282953125, 7.12700243751507901763289643588, 8.447837726563350907614761235554, 8.522861296168303217599352412099, 9.182042293678520906078410111145, 9.942438967164571860061575858903, 10.94942971943725818731871560682, 11.12157962911757596684358503736, 11.52443396620969954296692261123

Graph of the $Z$-function along the critical line