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Properties

Label 2-40e2-16.5-c1-0-8
Degree $2$
Conductor $1600$
Sign $0.382 - 0.923i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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  + (−2.15 + 2.15i)3-s − 2.31i·7-s − 6.31i·9-s + (−3.15 − 3.15i)11-s + (−4.31 + 4.31i)13-s + 1.31·17-s + (0.158 − 0.158i)19-s + (5 + 5i)21-s − 0.316i·23-s + (7.15 + 7.15i)27-s + (2 − 2i)29-s + 2.31·31-s + 13.6·33-s + (0.683 + 0.683i)37-s − 18.6i·39-s + ⋯
L(s)  = 1  + (−1.24 + 1.24i)3-s − 0.875i·7-s − 2.10i·9-s + (−0.952 − 0.952i)11-s + (−1.19 + 1.19i)13-s + 0.319·17-s + (0.0363 − 0.0363i)19-s + (1.09 + 1.09i)21-s − 0.0660i·23-s + (1.37 + 1.37i)27-s + (0.371 − 0.371i)29-s + 0.416·31-s + 2.37·33-s + (0.112 + 0.112i)37-s − 2.98i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7448149850\)
\(L(\frac12)\) \(\approx\) \(0.7448149850\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (2.15 - 2.15i)T - 3iT^{2} \)
7 \( 1 + 2.31iT - 7T^{2} \)
11 \( 1 + (3.15 + 3.15i)T + 11iT^{2} \)
13 \( 1 + (4.31 - 4.31i)T - 13iT^{2} \)
17 \( 1 - 1.31T + 17T^{2} \)
19 \( 1 + (-0.158 + 0.158i)T - 19iT^{2} \)
23 \( 1 + 0.316iT - 23T^{2} \)
29 \( 1 + (-2 + 2i)T - 29iT^{2} \)
31 \( 1 - 2.31T + 31T^{2} \)
37 \( 1 + (-0.683 - 0.683i)T + 37iT^{2} \)
41 \( 1 - 5iT - 41T^{2} \)
43 \( 1 + (-7.63 - 7.63i)T + 43iT^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (3.31 + 3.31i)T + 53iT^{2} \)
59 \( 1 + (-1.31 - 1.31i)T + 59iT^{2} \)
61 \( 1 + (-9.63 + 9.63i)T - 61iT^{2} \)
67 \( 1 + (9.15 - 9.15i)T - 67iT^{2} \)
71 \( 1 - 8.63iT - 71T^{2} \)
73 \( 1 - 6.68iT - 73T^{2} \)
79 \( 1 + 4.31T + 79T^{2} \)
83 \( 1 + (-7.15 + 7.15i)T - 83iT^{2} \)
89 \( 1 - 3.94iT - 89T^{2} \)
97 \( 1 + 6.63T + 97T^{2} \)
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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

−9.883019862293209974193747894489, −9.049268182129367501872402773732, −7.919100225552499923055426656151, −6.99306984029585002713909596574, −6.14340612147722856785704049096, −5.33628574601262472958724163697, −4.57205598899801062307428613661, −3.98707257861592397247559777689, −2.76021430837972293409226783409, −0.73080639112088721636148502379, 0.53390163103972747278135730645, 2.04783949570786613810893349096, 2.74789839054856825815862546200, 4.66775943924797378787877386985, 5.47878770274666310305277103694, 5.76236224648033828942621041757, 6.98572872660362309359265368566, 7.48704149673283759775075196555, 8.111569611119314836643051229523, 9.286691928895444772668376072397

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