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Properties

Label 2-3640-3640.1299-c0-0-7
Degree $2$
Conductor $3640$
Sign $0.281 + 0.959i$
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 + (0.984 − 0.173i)5-s − 1.28·6-s + (0.642 + 0.766i)7-s − 0.999i·8-s + (0.326 + 0.565i)9-s + (0.766 − 0.642i)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 + (0.984 − 0.173i)5-s − 1.28·6-s + (0.642 + 0.766i)7-s − 0.999i·8-s + (0.326 + 0.565i)9-s + (0.766 − 0.642i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $0.281 + 0.959i$
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: \((2,\ 3640,\ (\ :0),\ 0.281 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.991583459\)
\(L(\frac12)\) \(\approx\) \(1.991583459\)
\(L(1)\) 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 \( 1 + (-0.984 + 0.173i)T \)
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

−8.639816148394269614799206892156, −7.61039969389864556790064454006, −6.61947763855979312286132365592, −6.12925676972683223406023159954, −5.54420124705199444917119389704, −5.08414954582380885825164036352, −4.10379844113965780259965574389, −2.87021888755298669127414137263, −1.79261210104086322959193240318, −1.31868520695275247231880660687, 1.28878370398719845507450945999, 2.78123020297720633302462224967, 3.57154199213024726555372040878, 4.71808410293822143320851473028, 5.19288383318145058072710846038, 5.60236896274700628772753826799, 6.41095954752121587037846932467, 7.21164517121223301238635081139, 7.85522665016876257075200127860, 8.768241959320860964919990877745

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