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Properties

Label 2-40e2-16.13-c1-0-34
Degree $2$
Conductor $1600$
Sign $-0.997 - 0.0649i$
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  + (−0.720 − 0.720i)3-s − 4.02i·7-s − 1.96i·9-s + (0.646 − 0.646i)11-s + (−4.91 − 4.91i)13-s + 2.70·17-s + (0.438 + 0.438i)19-s + (−2.90 + 2.90i)21-s + 3.60i·23-s + (−3.57 + 3.57i)27-s + (2.00 + 2.00i)29-s − 4.30·31-s − 0.932·33-s + (0.743 − 0.743i)37-s + 7.08i·39-s + ⋯
L(s)  = 1  + (−0.416 − 0.416i)3-s − 1.52i·7-s − 0.653i·9-s + (0.195 − 0.195i)11-s + (−1.36 − 1.36i)13-s + 0.656·17-s + (0.100 + 0.100i)19-s + (−0.633 + 0.633i)21-s + 0.750i·23-s + (−0.688 + 0.688i)27-s + (0.373 + 0.373i)29-s − 0.774·31-s − 0.162·33-s + (0.122 − 0.122i)37-s + 1.13i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.997 - 0.0649i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.997 - 0.0649i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8370564753\)
\(L(\frac12)\) \(\approx\) \(0.8370564753\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.720 + 0.720i)T + 3iT^{2} \)
7 \( 1 + 4.02iT - 7T^{2} \)
11 \( 1 + (-0.646 + 0.646i)T - 11iT^{2} \)
13 \( 1 + (4.91 + 4.91i)T + 13iT^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
19 \( 1 + (-0.438 - 0.438i)T + 19iT^{2} \)
23 \( 1 - 3.60iT - 23T^{2} \)
29 \( 1 + (-2.00 - 2.00i)T + 29iT^{2} \)
31 \( 1 + 4.30T + 31T^{2} \)
37 \( 1 + (-0.743 + 0.743i)T - 37iT^{2} \)
41 \( 1 + 0.603iT - 41T^{2} \)
43 \( 1 + (5.03 - 5.03i)T - 43iT^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + (4.07 - 4.07i)T - 53iT^{2} \)
59 \( 1 + (1.22 - 1.22i)T - 59iT^{2} \)
61 \( 1 + (6.98 + 6.98i)T + 61iT^{2} \)
67 \( 1 + (-5.24 - 5.24i)T + 67iT^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 - 1.30iT - 73T^{2} \)
79 \( 1 + 0.611T + 79T^{2} \)
83 \( 1 + (-1.29 - 1.29i)T + 83iT^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−9.146855777730934759431400991244, −7.75234786717105095512298612191, −7.53443371560669562098697471371, −6.69584438046729845379071729895, −5.75461255066934238448194567147, −4.92915357716810144501193741758, −3.80646910439617594502006178855, −3.04405651429855429970309522319, −1.32406859020169411618368548510, −0.35278889753216687489216346938, 1.98000612207346402807359961596, 2.64900754737663091050818613999, 4.15145550972907772812348289042, 4.99031069101172624065913930146, 5.54342388700061290840121854726, 6.51480184519065605897318687962, 7.39749171012618882034793797802, 8.335405643688226272816151716219, 9.173032897861009099481008411489, 9.694053658797221699264246939673

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