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Properties

Label 2-40e2-80.67-c1-0-26
Degree $2$
Conductor $1600$
Sign $-0.834 + 0.551i$
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  − 1.96i·3-s + (−1.60 − 1.60i)7-s − 0.851·9-s + (−0.754 − 0.754i)11-s + 5.94·13-s + (−1.95 − 1.95i)17-s + (0.780 + 0.780i)19-s + (−3.14 + 3.14i)21-s + (4.93 − 4.93i)23-s − 4.21i·27-s + (1.44 − 1.44i)29-s − 3.60i·31-s + (−1.48 + 1.48i)33-s − 10.2·37-s − 11.6i·39-s + ⋯
L(s)  = 1  − 1.13i·3-s + (−0.605 − 0.605i)7-s − 0.283·9-s + (−0.227 − 0.227i)11-s + 1.64·13-s + (−0.474 − 0.474i)17-s + (0.179 + 0.179i)19-s + (−0.686 + 0.686i)21-s + (1.02 − 1.02i)23-s − 0.811i·27-s + (0.268 − 0.268i)29-s − 0.648i·31-s + (−0.257 + 0.257i)33-s − 1.68·37-s − 1.86i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.834 + 0.551i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.834 + 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.428590966\)
\(L(\frac12)\) \(\approx\) \(1.428590966\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.96iT - 3T^{2} \)
7 \( 1 + (1.60 + 1.60i)T + 7iT^{2} \)
11 \( 1 + (0.754 + 0.754i)T + 11iT^{2} \)
13 \( 1 - 5.94T + 13T^{2} \)
17 \( 1 + (1.95 + 1.95i)T + 17iT^{2} \)
19 \( 1 + (-0.780 - 0.780i)T + 19iT^{2} \)
23 \( 1 + (-4.93 + 4.93i)T - 23iT^{2} \)
29 \( 1 + (-1.44 + 1.44i)T - 29iT^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 6.93iT - 41T^{2} \)
43 \( 1 + 9.91T + 43T^{2} \)
47 \( 1 + (0.104 - 0.104i)T - 47iT^{2} \)
53 \( 1 + 4.03iT - 53T^{2} \)
59 \( 1 + (3.46 - 3.46i)T - 59iT^{2} \)
61 \( 1 + (-0.680 - 0.680i)T + 61iT^{2} \)
67 \( 1 + 9.04T + 67T^{2} \)
71 \( 1 - 3.64T + 71T^{2} \)
73 \( 1 + (2.94 + 2.94i)T + 73iT^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 4.23iT - 83T^{2} \)
89 \( 1 + 0.0426T + 89T^{2} \)
97 \( 1 + (-1.91 - 1.91i)T + 97iT^{2} \)
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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

−8.842588309121049856077043042773, −8.272906868314518039553731198035, −7.36655630504770118958401396417, −6.59909782189087050929261696914, −6.25615077017719247237310485205, −4.99187470974723487412786428522, −3.84545770733303149499535913819, −2.93180802023572637165550131577, −1.62419300883651015539050675880, −0.57797423577447640087525095608, 1.59823062899010311755647894020, 3.20340679698922588352446900907, 3.65581215135201668518887203123, 4.78958918353848610360363077233, 5.50948358899831651985149172223, 6.40947700832093448564522004441, 7.24882714013017642903663088753, 8.593698514550227160813441907101, 8.916459142548154219619741669964, 9.685257447813005340910662061381

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