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Properties

Label 2-40e2-80.69-c1-0-23
Degree $2$
Conductor $1600$
Sign $-0.331 + 0.943i$
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  + (−1.09 − 1.09i)3-s − 0.973·7-s − 0.616i·9-s + (−1.40 − 1.40i)11-s + (4.60 + 4.60i)13-s + 0.490i·17-s + (4.54 − 4.54i)19-s + (1.06 + 1.06i)21-s + 1.94·23-s + (−3.94 + 3.94i)27-s + (3.74 − 3.74i)29-s − 4.29·31-s + 3.07i·33-s + (−4.55 + 4.55i)37-s − 10.0i·39-s + ⋯
L(s)  = 1  + (−0.630 − 0.630i)3-s − 0.368·7-s − 0.205i·9-s + (−0.424 − 0.424i)11-s + (1.27 + 1.27i)13-s + 0.118i·17-s + (1.04 − 1.04i)19-s + (0.231 + 0.231i)21-s + 0.405·23-s + (−0.759 + 0.759i)27-s + (0.695 − 0.695i)29-s − 0.770·31-s + 0.535i·33-s + (−0.748 + 0.748i)37-s − 1.60i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.331 + 0.943i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.087924274\)
\(L(\frac12)\) \(\approx\) \(1.087924274\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 + 1.09i)T + 3iT^{2} \)
7 \( 1 + 0.973T + 7T^{2} \)
11 \( 1 + (1.40 + 1.40i)T + 11iT^{2} \)
13 \( 1 + (-4.60 - 4.60i)T + 13iT^{2} \)
17 \( 1 - 0.490iT - 17T^{2} \)
19 \( 1 + (-4.54 + 4.54i)T - 19iT^{2} \)
23 \( 1 - 1.94T + 23T^{2} \)
29 \( 1 + (-3.74 + 3.74i)T - 29iT^{2} \)
31 \( 1 + 4.29T + 31T^{2} \)
37 \( 1 + (4.55 - 4.55i)T - 37iT^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (-1.79 + 1.79i)T - 43iT^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + (-5.61 + 5.61i)T - 53iT^{2} \)
59 \( 1 + (-8.44 - 8.44i)T + 59iT^{2} \)
61 \( 1 + (-3.01 + 3.01i)T - 61iT^{2} \)
67 \( 1 + (7.07 + 7.07i)T + 67iT^{2} \)
71 \( 1 + 0.897iT - 71T^{2} \)
73 \( 1 + 9.71T + 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + (-0.815 - 0.815i)T + 83iT^{2} \)
89 \( 1 + 1.12iT - 89T^{2} \)
97 \( 1 + 7.54iT - 97T^{2} \)
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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

−8.992477887210818029045122741971, −8.563633987381283309829571663923, −7.19570069723975969818461023261, −6.84033472605764044924060597080, −5.97721227320232144500809446196, −5.27686004135150402490369895034, −4.03031854820902231228823717563, −3.13856456486660729029391328566, −1.71710988995705557045789812862, −0.51936673615164126059253469438, 1.23752913406789116194745354591, 2.88666675471376490006339571695, 3.72208646766753569880693627977, 4.82164245580513447149158191806, 5.56484285958928732016530884397, 6.12022346001002451337517259224, 7.35188872556495518987643698920, 8.017094456049415779840048920757, 8.909680816035273220402764141756, 9.949537592989672355986379891056

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