Properties

Label 2-20e2-5.2-c4-0-18
Degree $2$
Conductor $400$
Sign $0.850 + 0.525i$
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  + (1 − i)3-s + (−19 − 19i)7-s + 79i·9-s − 202·11-s + (99 − 99i)13-s + (239 + 239i)17-s − 40i·19-s − 38·21-s + (541 − 541i)23-s + (160 + 160i)27-s − 200i·29-s + 758·31-s + (−202 + 202i)33-s + (−141 − 141i)37-s − 198i·39-s + ⋯
L(s)  = 1  + (0.111 − 0.111i)3-s + (−0.387 − 0.387i)7-s + 0.975i·9-s − 1.66·11-s + (0.585 − 0.585i)13-s + (0.826 + 0.826i)17-s − 0.110i·19-s − 0.0861·21-s + (1.02 − 1.02i)23-s + (0.219 + 0.219i)27-s − 0.237i·29-s + 0.788·31-s + (−0.185 + 0.185i)33-s + (−0.102 − 0.102i)37-s − 0.130i·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.850 + 0.525i$
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: \((2,\ 400,\ (\ :2),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.737262896\)
\(L(\frac12)\) \(\approx\) \(1.737262896\)
\(L(3)\) 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 \( 1 + (-1 + i)T - 81iT^{2} \)
7 \( 1 + (19 + 19i)T + 2.40e3iT^{2} \)
11 \( 1 + 202T + 1.46e4T^{2} \)
13 \( 1 + (-99 + 99i)T - 2.85e4iT^{2} \)
17 \( 1 + (-239 - 239i)T + 8.35e4iT^{2} \)
19 \( 1 + 40iT - 1.30e5T^{2} \)
23 \( 1 + (-541 + 541i)T - 2.79e5iT^{2} \)
29 \( 1 + 200iT - 7.07e5T^{2} \)
31 \( 1 - 758T + 9.23e5T^{2} \)
37 \( 1 + (141 + 141i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.04e3T + 2.82e6T^{2} \)
43 \( 1 + (759 - 759i)T - 3.41e6iT^{2} \)
47 \( 1 + (459 + 459i)T + 4.87e6iT^{2} \)
53 \( 1 + (-1.81e3 + 1.81e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 4.60e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.08e3T + 1.38e7T^{2} \)
67 \( 1 + (-5.08e3 - 5.08e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 3.47e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.47e3 + 3.47e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 7.68e3iT - 3.89e7T^{2} \)
83 \( 1 + (-6.08e3 + 6.08e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 5.68e3iT - 6.27e7T^{2} \)
97 \( 1 + (561 + 561i)T + 8.85e7iT^{2} \)
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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

−10.49158500194937339125995954296, −9.979919091010681488928090983602, −8.377419750323141452871688536595, −7.995502519492353190216280757955, −6.89599229084264690259156278567, −5.65813194700031470701462559488, −4.82073919989512673996079293970, −3.35580476666520696700716649448, −2.31091479658257957665059452829, −0.64663781710665956281077495271, 0.886544906324745306484897915128, 2.64987863341247488337536180758, 3.53151697540289938751978429173, 4.99533928209570691286239003331, 5.89096410447756101411900770840, 6.99597750571766061785841767432, 7.970514844440842449184831378330, 9.051672661225350123619609264332, 9.708005266849248212602686325765, 10.68689138663013968532059969094

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