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Properties

Label 2-40e2-40.3-c1-0-11
Degree $2$
Conductor $1600$
Sign $-0.727 - 0.685i$
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  + (2.18 + 2.18i)3-s + (−1.79 − 1.79i)7-s + 6.58i·9-s + 0.913·11-s + (−1.73 + 1.73i)13-s + (−3 + 3i)17-s + 3.46i·19-s − 7.84i·21-s + (−3.79 + 3.79i)23-s + (−7.84 + 7.84i)27-s + 5.29·29-s + 7.58i·31-s + (1.99 + 1.99i)33-s + (−5.19 − 5.19i)37-s − 7.58·39-s + ⋯
L(s)  = 1  + (1.26 + 1.26i)3-s + (−0.677 − 0.677i)7-s + 2.19i·9-s + 0.275·11-s + (−0.480 + 0.480i)13-s + (−0.727 + 0.727i)17-s + 0.794i·19-s − 1.71i·21-s + (−0.790 + 0.790i)23-s + (−1.50 + 1.50i)27-s + 0.982·29-s + 1.36i·31-s + (0.348 + 0.348i)33-s + (−0.854 − 0.854i)37-s − 1.21·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.727 - 0.685i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.727 - 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.972147332\)
\(L(\frac12)\) \(\approx\) \(1.972147332\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 2.18i)T + 3iT^{2} \)
7 \( 1 + (1.79 + 1.79i)T + 7iT^{2} \)
11 \( 1 - 0.913T + 11T^{2} \)
13 \( 1 + (1.73 - 1.73i)T - 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + (3.79 - 3.79i)T - 23iT^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 - 7.58iT - 31T^{2} \)
37 \( 1 + (5.19 + 5.19i)T + 37iT^{2} \)
41 \( 1 - 1.58T + 41T^{2} \)
43 \( 1 + (-0.361 - 0.361i)T + 43iT^{2} \)
47 \( 1 + (-3.79 - 3.79i)T + 47iT^{2} \)
53 \( 1 + (-2.64 + 2.64i)T - 53iT^{2} \)
59 \( 1 - 5.29iT - 59T^{2} \)
61 \( 1 - 6.20iT - 61T^{2} \)
67 \( 1 + (0.361 - 0.361i)T - 67iT^{2} \)
71 \( 1 - 4.41iT - 71T^{2} \)
73 \( 1 + (-8.58 - 8.58i)T + 73iT^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + (-3.10 - 3.10i)T + 83iT^{2} \)
89 \( 1 + 15.1iT - 89T^{2} \)
97 \( 1 + (-8.58 + 8.58i)T - 97iT^{2} \)
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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

−9.728382120215094001341538674089, −8.964473661178827998658718271996, −8.386683739132600607113783980557, −7.47073805936899220178967510872, −6.61658050874274459835521856127, −5.42076989601649586303408966000, −4.28801762862615721521363946868, −3.87702192037804588662254131584, −3.01485099873229201490784399340, −1.87424781492639136427650538632, 0.61095944695128959076990092576, 2.24650422380676784134967462849, 2.64002934985062520897712609849, 3.66406965006767386993784660837, 4.95507264789396189869087507119, 6.33954456270885379243637060984, 6.64559181242840919985688277878, 7.64158293508869080125687689823, 8.235409793740451254297046833008, 9.118145262082130838688614479319

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