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Properties

Label 2-20e2-5.2-c4-0-1
Degree $2$
Conductor $400$
Sign $-0.525 + 0.850i$
Analytic cond. $41.3479$
Root an. cond. $6.43023$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.8 + 11.8i)3-s + (19.8 + 19.8i)7-s − 197. i·9-s + 184.·11-s + (−189. + 189. i)13-s + (−34.2 − 34.2i)17-s + 474. i·19-s − 467.·21-s + (−409. + 409. i)23-s + (1.38e3 + 1.38e3i)27-s + 349. i·29-s − 1.52e3·31-s + (−2.18e3 + 2.18e3i)33-s + (−85.2 − 85.2i)37-s − 4.48e3i·39-s + ⋯
L(s)  = 1  + (−1.31 + 1.31i)3-s + (0.404 + 0.404i)7-s − 2.44i·9-s + 1.52·11-s + (−1.12 + 1.12i)13-s + (−0.118 − 0.118i)17-s + 1.31i·19-s − 1.06·21-s + (−0.773 + 0.773i)23-s + (1.89 + 1.89i)27-s + 0.415i·29-s − 1.58·31-s + (−2.00 + 2.00i)33-s + (−0.0622 − 0.0622i)37-s − 2.94i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(41.3479\)
Root analytic conductor: \(6.43023\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :2),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4479250802\)
\(L(\frac12)\) \(\approx\) \(0.4479250802\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (11.8 - 11.8i)T - 81iT^{2} \)
7 \( 1 + (-19.8 - 19.8i)T + 2.40e3iT^{2} \)
11 \( 1 - 184.T + 1.46e4T^{2} \)
13 \( 1 + (189. - 189. i)T - 2.85e4iT^{2} \)
17 \( 1 + (34.2 + 34.2i)T + 8.35e4iT^{2} \)
19 \( 1 - 474. iT - 1.30e5T^{2} \)
23 \( 1 + (409. - 409. i)T - 2.79e5iT^{2} \)
29 \( 1 - 349. iT - 7.07e5T^{2} \)
31 \( 1 + 1.52e3T + 9.23e5T^{2} \)
37 \( 1 + (85.2 + 85.2i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.30e3T + 2.82e6T^{2} \)
43 \( 1 + (-393. + 393. i)T - 3.41e6iT^{2} \)
47 \( 1 + (-2.24e3 - 2.24e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-989. + 989. i)T - 7.89e6iT^{2} \)
59 \( 1 - 558. iT - 1.21e7T^{2} \)
61 \( 1 + 3.48e3T + 1.38e7T^{2} \)
67 \( 1 + (2.67e3 + 2.67e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 3.96e3T + 2.54e7T^{2} \)
73 \( 1 + (169. - 169. i)T - 2.83e7iT^{2} \)
79 \( 1 - 4.80e3iT - 3.89e7T^{2} \)
83 \( 1 + (-3.27e3 + 3.27e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 8.88e3iT - 6.27e7T^{2} \)
97 \( 1 + (8.04e3 + 8.04e3i)T + 8.85e7iT^{2} \)
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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

−11.35585947828238597840696592306, −10.37678331295691482116875668622, −9.448278336875816870406585058060, −9.054802111058807613637375883914, −7.31110159447919168803201196189, −6.21707426416967691457615763742, −5.47146884374848515839876647310, −4.39980830041892592087899420875, −3.75960822459243467716151809188, −1.62102422920841957325740218090, 0.17301657358078492920089957064, 1.09873585277115317466800868535, 2.34262661673408208814262187458, 4.33706809800483150994673716193, 5.38434516269740065480258636664, 6.30964042999914157315583522878, 7.16379186602550542405384469036, 7.75410527214562413971104776833, 9.079208708352819959082243372086, 10.42608312751507253596449551627

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