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Properties

Label 2-40e2-16.5-c1-0-16
Degree $2$
Conductor $1600$
Sign $0.986 - 0.163i$
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  + (1.03 − 1.03i)3-s + 1.49i·7-s + 0.836i·9-s + (−0.423 − 0.423i)11-s + (−1.85 + 1.85i)13-s + 6.50·17-s + (1.75 − 1.75i)19-s + (1.55 + 1.55i)21-s − 7.19i·23-s + (3.99 + 3.99i)27-s + (−6.57 + 6.57i)29-s + 6.75·31-s − 0.880·33-s + (1.95 + 1.95i)37-s + 3.86i·39-s + ⋯
L(s)  = 1  + (0.600 − 0.600i)3-s + 0.565i·7-s + 0.278i·9-s + (−0.127 − 0.127i)11-s + (−0.515 + 0.515i)13-s + 1.57·17-s + (0.403 − 0.403i)19-s + (0.339 + 0.339i)21-s − 1.49i·23-s + (0.767 + 0.767i)27-s + (−1.22 + 1.22i)29-s + 1.21·31-s − 0.153·33-s + (0.321 + 0.321i)37-s + 0.618i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.986 - 0.163i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.986 - 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.136462034\)
\(L(\frac12)\) \(\approx\) \(2.136462034\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 + 1.03i)T - 3iT^{2} \)
7 \( 1 - 1.49iT - 7T^{2} \)
11 \( 1 + (0.423 + 0.423i)T + 11iT^{2} \)
13 \( 1 + (1.85 - 1.85i)T - 13iT^{2} \)
17 \( 1 - 6.50T + 17T^{2} \)
19 \( 1 + (-1.75 + 1.75i)T - 19iT^{2} \)
23 \( 1 + 7.19iT - 23T^{2} \)
29 \( 1 + (6.57 - 6.57i)T - 29iT^{2} \)
31 \( 1 - 6.75T + 31T^{2} \)
37 \( 1 + (-1.95 - 1.95i)T + 37iT^{2} \)
41 \( 1 - 7.70iT - 41T^{2} \)
43 \( 1 + (-6.13 - 6.13i)T + 43iT^{2} \)
47 \( 1 - 6.65T + 47T^{2} \)
53 \( 1 + (-5.29 - 5.29i)T + 53iT^{2} \)
59 \( 1 + (5.91 + 5.91i)T + 59iT^{2} \)
61 \( 1 + (1.43 - 1.43i)T - 61iT^{2} \)
67 \( 1 + (-6.35 + 6.35i)T - 67iT^{2} \)
71 \( 1 - 4.08iT - 71T^{2} \)
73 \( 1 + 2.43iT - 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + (2.81 - 2.81i)T - 83iT^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−9.279499643983910139832536592967, −8.577108756882685578430734263019, −7.80472589445050770979163730262, −7.24422232402983105701835594970, −6.25236445797374800523559049294, −5.33038480414116251443707252579, −4.47929319054262610922528083953, −3.07411135420637881352495045481, −2.46325247833040978333437853352, −1.22704177544947225387050449276, 0.910857408873039630948379708474, 2.50058294593040282366239679534, 3.57692070871076218051118847601, 4.02921369769952493256625418167, 5.32748601278014462006907293422, 5.93430429135434927858721952512, 7.42418782063202211586484688539, 7.59636121874798399047726197260, 8.680673458383612051833733738578, 9.591571089725987493064895489137

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