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Properties

Label 2-40e2-16.13-c1-0-12
Degree $2$
Conductor $1600$
Sign $-0.382 - 0.923i$
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  + (2.15 + 2.15i)3-s − 2.31i·7-s + 6.31i·9-s + (−3.15 + 3.15i)11-s + (4.31 + 4.31i)13-s − 1.31·17-s + (0.158 + 0.158i)19-s + (5 − 5i)21-s − 0.316i·23-s + (−7.15 + 7.15i)27-s + (2 + 2i)29-s + 2.31·31-s − 13.6·33-s + (−0.683 + 0.683i)37-s + 18.6i·39-s + ⋯
L(s)  = 1  + (1.24 + 1.24i)3-s − 0.875i·7-s + 2.10i·9-s + (−0.952 + 0.952i)11-s + (1.19 + 1.19i)13-s − 0.319·17-s + (0.0363 + 0.0363i)19-s + (1.09 − 1.09i)21-s − 0.0660i·23-s + (−1.37 + 1.37i)27-s + (0.371 + 0.371i)29-s + 0.416·31-s − 2.37·33-s + (−0.112 + 0.112i)37-s + 2.98i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.382 - 0.923i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.437104702\)
\(L(\frac12)\) \(\approx\) \(2.437104702\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 - 2.15i)T + 3iT^{2} \)
7 \( 1 + 2.31iT - 7T^{2} \)
11 \( 1 + (3.15 - 3.15i)T - 11iT^{2} \)
13 \( 1 + (-4.31 - 4.31i)T + 13iT^{2} \)
17 \( 1 + 1.31T + 17T^{2} \)
19 \( 1 + (-0.158 - 0.158i)T + 19iT^{2} \)
23 \( 1 + 0.316iT - 23T^{2} \)
29 \( 1 + (-2 - 2i)T + 29iT^{2} \)
31 \( 1 - 2.31T + 31T^{2} \)
37 \( 1 + (0.683 - 0.683i)T - 37iT^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 + (7.63 - 7.63i)T - 43iT^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (-3.31 + 3.31i)T - 53iT^{2} \)
59 \( 1 + (-1.31 + 1.31i)T - 59iT^{2} \)
61 \( 1 + (-9.63 - 9.63i)T + 61iT^{2} \)
67 \( 1 + (-9.15 - 9.15i)T + 67iT^{2} \)
71 \( 1 + 8.63iT - 71T^{2} \)
73 \( 1 - 6.68iT - 73T^{2} \)
79 \( 1 + 4.31T + 79T^{2} \)
83 \( 1 + (7.15 + 7.15i)T + 83iT^{2} \)
89 \( 1 + 3.94iT - 89T^{2} \)
97 \( 1 - 6.63T + 97T^{2} \)
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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

−9.831268378000621239166509816122, −8.781374383616868492056825842413, −8.365959147492488045216193215691, −7.41706877624103409564440434780, −6.62514438861370055891868707804, −5.17657053403341719945827495543, −4.37925767753785099134088072973, −3.86643420466276872888524017374, −2.86508287219055308500493727555, −1.77260668700344983357651641603, 0.806346660310566987761366925076, 2.11656108032855161315923276747, 2.94781378657389424228346027498, 3.55774711321549254641807374276, 5.27427660252145941459382692496, 6.05235876050997606364239793369, 6.80041571069523751460433435231, 7.932212460904680029787595114674, 8.320911008770646419945463414612, 8.702010053341982647950095966861

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