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Properties

Label 2-40e2-80.67-c1-0-18
Degree $2$
Conductor $1600$
Sign $0.625 + 0.779i$
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  − 0.619i·3-s + (−1.82 − 1.82i)7-s + 2.61·9-s + (0.567 + 0.567i)11-s − 2.78·13-s + (3.65 + 3.65i)17-s + (4.51 + 4.51i)19-s + (−1.12 + 1.12i)21-s + (2.15 − 2.15i)23-s − 3.47i·27-s + (3.20 − 3.20i)29-s − 3.54i·31-s + (0.351 − 0.351i)33-s + 5.22·37-s + 1.72i·39-s + ⋯
L(s)  = 1  − 0.357i·3-s + (−0.689 − 0.689i)7-s + 0.872·9-s + (0.171 + 0.171i)11-s − 0.773·13-s + (0.885 + 0.885i)17-s + (1.03 + 1.03i)19-s + (−0.246 + 0.246i)21-s + (0.449 − 0.449i)23-s − 0.669i·27-s + (0.594 − 0.594i)29-s − 0.635i·31-s + (0.0611 − 0.0611i)33-s + 0.858·37-s + 0.276i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.625 + 0.779i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.625 + 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.690045239\)
\(L(\frac12)\) \(\approx\) \(1.690045239\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.619iT - 3T^{2} \)
7 \( 1 + (1.82 + 1.82i)T + 7iT^{2} \)
11 \( 1 + (-0.567 - 0.567i)T + 11iT^{2} \)
13 \( 1 + 2.78T + 13T^{2} \)
17 \( 1 + (-3.65 - 3.65i)T + 17iT^{2} \)
19 \( 1 + (-4.51 - 4.51i)T + 19iT^{2} \)
23 \( 1 + (-2.15 + 2.15i)T - 23iT^{2} \)
29 \( 1 + (-3.20 + 3.20i)T - 29iT^{2} \)
31 \( 1 + 3.54iT - 31T^{2} \)
37 \( 1 - 5.22T + 37T^{2} \)
41 \( 1 + 8.76iT - 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (-3.22 + 3.22i)T - 47iT^{2} \)
53 \( 1 + 12.8iT - 53T^{2} \)
59 \( 1 + (3.79 - 3.79i)T - 59iT^{2} \)
61 \( 1 + (-6.63 - 6.63i)T + 61iT^{2} \)
67 \( 1 - 7.78T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + (1.34 + 1.34i)T + 73iT^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + 0.391iT - 83T^{2} \)
89 \( 1 - 18.0T + 89T^{2} \)
97 \( 1 + (6.43 + 6.43i)T + 97iT^{2} \)
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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

−9.591310219847764510697532994800, −8.319360795978639243693243925488, −7.58214941660758351645133041904, −6.98339570383132911783253208848, −6.20232216326550924918127170486, −5.19275111729491235986414003641, −4.09974497101525042451697353435, −3.40248678546078201676946796571, −2.02141755206620328525639862821, −0.812478562741447178995921788857, 1.12434517669904170767226444366, 2.75597836037670781382382628789, 3.34980619070430008671435437169, 4.74946750415435401441312949213, 5.19078658913218898987794655651, 6.35693618581237584897120514479, 7.12969202481778227573265218983, 7.81300683164169166587296471102, 9.066890974290859682949777568451, 9.513798179971853184230259316721

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