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Properties

Label 2-40e2-16.5-c1-0-21
Degree $2$
Conductor $1600$
Sign $0.725 + 0.687i$
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  + (−1.66 + 1.66i)3-s + 2.89i·7-s − 2.53i·9-s + (−1.84 − 1.84i)11-s + (3.08 − 3.08i)13-s − 7.29·17-s + (1.23 − 1.23i)19-s + (−4.81 − 4.81i)21-s − 4.60i·23-s + (−0.772 − 0.772i)27-s + (4.24 − 4.24i)29-s − 2.06·31-s + 6.13·33-s + (1.17 + 1.17i)37-s + 10.2i·39-s + ⋯
L(s)  = 1  + (−0.960 + 0.960i)3-s + 1.09i·7-s − 0.845i·9-s + (−0.556 − 0.556i)11-s + (0.854 − 0.854i)13-s − 1.77·17-s + (0.283 − 0.283i)19-s + (−1.05 − 1.05i)21-s − 0.960i·23-s + (−0.148 − 0.148i)27-s + (0.788 − 0.788i)29-s − 0.370·31-s + 1.06·33-s + (0.193 + 0.193i)37-s + 1.64i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.725 + 0.687i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.725 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6357243822\)
\(L(\frac12)\) \(\approx\) \(0.6357243822\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 - 1.66i)T - 3iT^{2} \)
7 \( 1 - 2.89iT - 7T^{2} \)
11 \( 1 + (1.84 + 1.84i)T + 11iT^{2} \)
13 \( 1 + (-3.08 + 3.08i)T - 13iT^{2} \)
17 \( 1 + 7.29T + 17T^{2} \)
19 \( 1 + (-1.23 + 1.23i)T - 19iT^{2} \)
23 \( 1 + 4.60iT - 23T^{2} \)
29 \( 1 + (-4.24 + 4.24i)T - 29iT^{2} \)
31 \( 1 + 2.06T + 31T^{2} \)
37 \( 1 + (-1.17 - 1.17i)T + 37iT^{2} \)
41 \( 1 - 4.61iT - 41T^{2} \)
43 \( 1 + (-3.03 - 3.03i)T + 43iT^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + (2.73 + 2.73i)T + 53iT^{2} \)
59 \( 1 + (3.11 + 3.11i)T + 59iT^{2} \)
61 \( 1 + (-2.34 + 2.34i)T - 61iT^{2} \)
67 \( 1 + (-8.24 + 8.24i)T - 67iT^{2} \)
71 \( 1 - 3.25iT - 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 - 0.113T + 79T^{2} \)
83 \( 1 + (-9.76 + 9.76i)T - 83iT^{2} \)
89 \( 1 + 3.74iT - 89T^{2} \)
97 \( 1 - 13.9T + 97T^{2} \)
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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

−9.368568436299789236230810182518, −8.595544569140524988793453129495, −7.975509335892993742652212387750, −6.35799843170589529261715579258, −6.12196065258091623092945026787, −5.07548597902655547415805120588, −4.60765375523535899251874365777, −3.34625037381025477035901875532, −2.32013611511606535058047697929, −0.31729965873638083426994024406, 1.09948250765368499163253185177, 2.07104443206884888562458387018, 3.69472096177913353912555727782, 4.57316748898101601850384197797, 5.52750769361810520513775486552, 6.54340426020847544458262749717, 6.93869061965885125324023057534, 7.60446944636975197974347767384, 8.631572804447967944881588435170, 9.534336153292828810711881408255

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