Properties

Label 2-3640-3640.1299-c0-0-13
Degree $2$
Conductor $3640$
Sign $0.832 + 0.553i$
Analytic cond. $1.81659$
Root an. cond. $1.34781$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (1.11 + 0.642i)3-s + (0.499 − 0.866i)4-s + (0.642 − 0.766i)5-s + 1.28·6-s + (0.642 + 0.766i)7-s − 0.999i·8-s + (0.326 + 0.565i)9-s + (0.173 − 0.984i)10-s + (1.11 − 0.642i)12-s + i·13-s + (0.939 + 0.342i)14-s + (1.20 − 0.439i)15-s + (−0.5 − 0.866i)16-s + (−1.70 − 0.984i)17-s + (0.565 + 0.326i)18-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (1.11 + 0.642i)3-s + (0.499 − 0.866i)4-s + (0.642 − 0.766i)5-s + 1.28·6-s + (0.642 + 0.766i)7-s − 0.999i·8-s + (0.326 + 0.565i)9-s + (0.173 − 0.984i)10-s + (1.11 − 0.642i)12-s + i·13-s + (0.939 + 0.342i)14-s + (1.20 − 0.439i)15-s + (−0.5 − 0.866i)16-s + (−1.70 − 0.984i)17-s + (0.565 + 0.326i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.443737167\)
\(L(\frac12)\) \(\approx\) \(3.443737167\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.642 + 0.766i)T \)
7 \( 1 + (-0.642 - 0.766i)T \)
13 \( 1 - iT \)
good3 \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.32 - 0.766i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.684iT - T^{2} \)
47 \( 1 + (0.300 - 0.173i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.96T + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.994323986615914114054147566543, −8.263529233258500339511769490362, −6.97860813712941421047016158133, −6.31233048908912313491532764695, −5.18687718668973217609122716531, −4.80387325523212966418954571897, −4.11276245204479178983258217529, −3.07157360548803723127928548877, −2.25228183856220593164197670024, −1.66561102296930731723405124422, 1.93547240257327305875958737663, 2.27891096381898953825989990816, 3.40792952469189647150167663179, 3.95592340749042479672300415738, 5.07531866955763064060284918027, 5.88740993437270763709438244533, 6.73660586762038659457370753704, 7.25315066942754115172321832355, 7.88697033069473447411413452840, 8.475882486274791064362554802015

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