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Properties

Label 2-40e2-80.69-c1-0-5
Degree $2$
Conductor $1600$
Sign $-0.504 - 0.863i$
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 − 4.02·7-s − 1.96i·9-s + (0.646 + 0.646i)11-s + (4.91 + 4.91i)13-s + 2.70i·17-s + (−0.438 + 0.438i)19-s + (−2.90 − 2.90i)21-s − 3.60·23-s + (3.57 − 3.57i)27-s + (−2.00 + 2.00i)29-s − 4.30·31-s + 0.932i·33-s + (−0.743 + 0.743i)37-s + 7.08i·39-s + ⋯
L(s)  = 1  + (0.416 + 0.416i)3-s − 1.52·7-s − 0.653i·9-s + (0.195 + 0.195i)11-s + (1.36 + 1.36i)13-s + 0.656i·17-s + (−0.100 + 0.100i)19-s + (−0.633 − 0.633i)21-s − 0.750·23-s + (0.688 − 0.688i)27-s + (−0.373 + 0.373i)29-s − 0.774·31-s + 0.162i·33-s + (−0.122 + 0.122i)37-s + 1.13i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.504 - 0.863i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.504 - 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.141499271\)
\(L(\frac12)\) \(\approx\) \(1.141499271\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.02T + 7T^{2} \)
11 \( 1 + (-0.646 - 0.646i)T + 11iT^{2} \)
13 \( 1 + (-4.91 - 4.91i)T + 13iT^{2} \)
17 \( 1 - 2.70iT - 17T^{2} \)
19 \( 1 + (0.438 - 0.438i)T - 19iT^{2} \)
23 \( 1 + 3.60T + 23T^{2} \)
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 \( 1 - 10.8iT - 47T^{2} \)
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 \( 1 + 1.30T + 73T^{2} \)
79 \( 1 - 0.611T + 79T^{2} \)
83 \( 1 + (1.29 + 1.29i)T + 83iT^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 - 12.7iT - 97T^{2} \)
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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

−9.504506197012738670369457614371, −9.076354371820630599138009481667, −8.352671796024990825120956616163, −7.08920926302522546407974385033, −6.34592068122592449896763147517, −5.94909560627953416730294808061, −4.26807460911610582027050340792, −3.77893435578407487897538436322, −2.97625651909177001504706823332, −1.49957943792150658629553790068, 0.41540185773359720098701208793, 2.01077451197280624428871179051, 3.19905023635218841523190940573, 3.65002662297151585900155132422, 5.17610526449201428831546052716, 5.99262045460473019845782600971, 6.71401168559706959819478033146, 7.60557431958561636138817600697, 8.339329125102618665612030372214, 9.059117247261497360681975584849

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