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Properties

Label 2-40e2-80.43-c1-0-23
Degree $2$
Conductor $1600$
Sign $0.638 + 0.769i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.079919965\)
\(L(\frac12)\) \(\approx\) \(2.079919965\)
\(L(\frac{3}{2})\) 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

−9.205558455732301853253894232863, −8.420560082290998480165978593843, −7.69777496721985381255453162998, −6.86182500307488981680119053924, −6.30544295158651721567147669347, −4.99648726419942270802108055464, −4.32258711386347933613204298661, −3.35678942439148764143947079365, −1.89470825445229494531528455785, −0.973270910317105124972477301017, 1.30546956014929472264831052307, 2.40066388831285110117159467578, 3.72528001681509577745643761484, 4.47230082150674227365093719149, 5.39496009839716524812915864750, 6.15289421820676123129801266992, 7.27936853708205413141054174765, 7.86311081654607087612322926936, 8.936373156256896229666464344876, 9.365315447408555143379560735819

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