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Properties

Label 2-40e2-80.27-c1-0-13
Degree $2$
Conductor $1600$
Sign $0.753 - 0.657i$
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  − 0.496·3-s + (1.55 + 1.55i)7-s − 2.75·9-s + (4.19 − 4.19i)11-s + 5.09i·13-s + (−0.213 − 0.213i)17-s + (−0.844 + 0.844i)19-s + (−0.771 − 0.771i)21-s + (1.70 − 1.70i)23-s + 2.85·27-s + (2.24 + 2.24i)29-s − 0.818i·31-s + (−2.08 + 2.08i)33-s + 5.12i·37-s − 2.52i·39-s + ⋯
L(s)  = 1  − 0.286·3-s + (0.587 + 0.587i)7-s − 0.917·9-s + (1.26 − 1.26i)11-s + 1.41i·13-s + (−0.0517 − 0.0517i)17-s + (−0.193 + 0.193i)19-s + (−0.168 − 0.168i)21-s + (0.356 − 0.356i)23-s + 0.549·27-s + (0.417 + 0.417i)29-s − 0.146i·31-s + (−0.362 + 0.362i)33-s + 0.842i·37-s − 0.405i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.753 - 0.657i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.571772531\)
\(L(\frac12)\) \(\approx\) \(1.571772531\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.496T + 3T^{2} \)
7 \( 1 + (-1.55 - 1.55i)T + 7iT^{2} \)
11 \( 1 + (-4.19 + 4.19i)T - 11iT^{2} \)
13 \( 1 - 5.09iT - 13T^{2} \)
17 \( 1 + (0.213 + 0.213i)T + 17iT^{2} \)
19 \( 1 + (0.844 - 0.844i)T - 19iT^{2} \)
23 \( 1 + (-1.70 + 1.70i)T - 23iT^{2} \)
29 \( 1 + (-2.24 - 2.24i)T + 29iT^{2} \)
31 \( 1 + 0.818iT - 31T^{2} \)
37 \( 1 - 5.12iT - 37T^{2} \)
41 \( 1 - 3.34iT - 41T^{2} \)
43 \( 1 - 4.49iT - 43T^{2} \)
47 \( 1 + (-4.29 + 4.29i)T - 47iT^{2} \)
53 \( 1 - 1.00T + 53T^{2} \)
59 \( 1 + (-7.65 - 7.65i)T + 59iT^{2} \)
61 \( 1 + (1.90 - 1.90i)T - 61iT^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (2.70 + 2.70i)T + 73iT^{2} \)
79 \( 1 - 8.32T + 79T^{2} \)
83 \( 1 + 9.17T + 83T^{2} \)
89 \( 1 + 4.25T + 89T^{2} \)
97 \( 1 + (-7.15 - 7.15i)T + 97iT^{2} \)
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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

−9.168977286947261444795046522656, −8.783066536451189368779022328465, −8.190195333473550559533188396295, −6.83432280786488846246456270884, −6.27874664307296433163779142277, −5.49425737687090773292054106922, −4.53611742334095898341756326644, −3.55195599825564728912243005710, −2.41974233838896250860992877941, −1.13793770650957528359873610229, 0.76240990437425127701814325836, 2.09759476033844518519589828647, 3.37045339280076315826691347106, 4.35190770172947695120569072909, 5.17686540970242576410516028166, 6.03658212627629374231541449100, 6.97203366907916102425611689855, 7.67296628029045984877028379602, 8.508955781585193070388066102375, 9.320821556643424638024680174160

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