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Properties

Label 2-20e2-16.5-c1-0-16
Degree $2$
Conductor $400$
Sign $-0.0680 + 0.997i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.114 − 1.40i)2-s + (−1.42 + 1.42i)3-s + (−1.97 + 0.323i)4-s + (2.16 + 1.84i)6-s − 0.690i·7-s + (0.681 + 2.74i)8-s − 1.05i·9-s + (−3.06 − 3.06i)11-s + (2.34 − 3.26i)12-s + (2.33 − 2.33i)13-s + (−0.973 + 0.0791i)14-s + (3.79 − 1.27i)16-s + 5.28·17-s + (−1.48 + 0.120i)18-s + (5.38 − 5.38i)19-s + ⋯
L(s)  = 1  + (−0.0810 − 0.996i)2-s + (−0.821 + 0.821i)3-s + (−0.986 + 0.161i)4-s + (0.885 + 0.752i)6-s − 0.261i·7-s + (0.241 + 0.970i)8-s − 0.350i·9-s + (−0.922 − 0.922i)11-s + (0.678 − 0.943i)12-s + (0.648 − 0.648i)13-s + (−0.260 + 0.0211i)14-s + (0.947 − 0.318i)16-s + 1.28·17-s + (−0.349 + 0.0283i)18-s + (1.23 − 1.23i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0680 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0680 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.0680 + 0.997i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.0680 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.538091 - 0.576040i\)
\(L(\frac12)\) \(\approx\) \(0.538091 - 0.576040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.114 + 1.40i)T \)
5 \( 1 \)
good3 \( 1 + (1.42 - 1.42i)T - 3iT^{2} \)
7 \( 1 + 0.690iT - 7T^{2} \)
11 \( 1 + (3.06 + 3.06i)T + 11iT^{2} \)
13 \( 1 + (-2.33 + 2.33i)T - 13iT^{2} \)
17 \( 1 - 5.28T + 17T^{2} \)
19 \( 1 + (-5.38 + 5.38i)T - 19iT^{2} \)
23 \( 1 + 1.60iT - 23T^{2} \)
29 \( 1 + (-1.70 + 1.70i)T - 29iT^{2} \)
31 \( 1 + 4.69T + 31T^{2} \)
37 \( 1 + (7.89 + 7.89i)T + 37iT^{2} \)
41 \( 1 - 5.49iT - 41T^{2} \)
43 \( 1 + (-0.256 - 0.256i)T + 43iT^{2} \)
47 \( 1 - 4.60T + 47T^{2} \)
53 \( 1 + (-4.99 - 4.99i)T + 53iT^{2} \)
59 \( 1 + (-1.46 - 1.46i)T + 59iT^{2} \)
61 \( 1 + (-9.33 + 9.33i)T - 61iT^{2} \)
67 \( 1 + (-1.94 + 1.94i)T - 67iT^{2} \)
71 \( 1 - 2.32iT - 71T^{2} \)
73 \( 1 + 1.29iT - 73T^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + (7.30 - 7.30i)T - 83iT^{2} \)
89 \( 1 + 1.81iT - 89T^{2} \)
97 \( 1 + 5.27T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.90788904232555033259076784986, −10.43747862750309506854679805968, −9.597670022177050276890129709639, −8.512556629895108481351367276093, −7.52789915331829415370023152929, −5.57824500314671233431939784580, −5.27490852755335844363659490768, −3.90186958528449605050217470986, −2.89195919992046329885381520121, −0.68055648327532753740146349419, 1.37641565209357362711342998438, 3.69095231959127879979907935817, 5.32833743935718286769043261734, 5.69520622202601961360324321423, 6.95343390333775479054463458237, 7.46463316791008476663062238976, 8.479700951317189326215061029590, 9.664117880870959785695505266289, 10.42530900768471604822854225031, 11.89833314857917278910273797499

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