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Properties

Label 2-3640-3640.1299-c0-0-0
Degree $2$
Conductor $3640$
Sign $-0.971 - 0.235i$
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 + (−0.592 − 0.342i)3-s + (0.499 − 0.866i)4-s + (0.642 + 0.766i)5-s + 0.684·6-s + (−0.342 + 0.939i)7-s + 0.999i·8-s + (−0.266 − 0.460i)9-s + (−0.939 − 0.342i)10-s + (−0.592 + 0.342i)12-s − i·13-s + (−0.173 − 0.984i)14-s + (−0.118 − 0.673i)15-s + (−0.5 − 0.866i)16-s + (−1.11 − 0.642i)17-s + (0.460 + 0.266i)18-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.592 − 0.342i)3-s + (0.499 − 0.866i)4-s + (0.642 + 0.766i)5-s + 0.684·6-s + (−0.342 + 0.939i)7-s + 0.999i·8-s + (−0.266 − 0.460i)9-s + (−0.939 − 0.342i)10-s + (−0.592 + 0.342i)12-s − i·13-s + (−0.173 − 0.984i)14-s + (−0.118 − 0.673i)15-s + (−0.5 − 0.866i)16-s + (−1.11 − 0.642i)17-s + (0.460 + 0.266i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-0.971 - 0.235i$
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: \((2,\ 3640,\ (\ :0),\ -0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2638149397\)
\(L(\frac12)\) \(\approx\) \(0.2638149397\)
\(L(1)\) 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 \( 1 + (0.342 - 0.939i)T \)
13 \( 1 + iT \)
good3 \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.96iT - T^{2} \)
47 \( 1 + (-1.32 + 0.766i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + 1.28T + T^{2} \)
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

−9.009230333706587061047490689166, −8.488217017031724613748474774750, −7.38316422476542419679429295935, −6.74300318925718010348518074184, −6.30399291599263893455457095693, −5.54517183097235889545792472514, −5.08793620147015501932928209026, −3.22833384265688346013319982402, −2.57079870656955628012654639972, −1.45563525817321769960518901625, 0.21106438975938721875462475317, 1.67875735166755784632456215838, 2.37870285722090415434128120818, 3.94634968353854510063950403631, 4.27114383173717219095791865358, 5.40691147493779178469560151550, 6.22413060296426846557152155626, 6.98717197484463123163284161283, 7.71238193755425629208016969162, 8.725222821085937242903626172483

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