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Properties

Label 2-3640-3640.1299-c0-0-9
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 + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s − i·7-s + 0.999i·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (1.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s + (1.5 − 0.866i)19-s − 0.999i·20-s − 1.73·22-s + (−0.866 − 1.5i)23-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s − i·7-s + 0.999i·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (1.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s + (1.5 − 0.866i)19-s − 0.999i·20-s − 1.73·22-s + (−0.866 − 1.5i)23-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

\(L(\frac{1}{2})\) \(\approx\) \(1.129253410\)
\(L(\frac12)\) \(\approx\) \(1.129253410\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + iT \)
13 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + T^{2} \)
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

−8.802308343664741562639121066728, −8.028607332132899218417093762205, −6.99909141355611989092028318254, −6.48664426385386183591837300869, −6.10646397177493886992540910198, −4.89321976785385308020954365239, −4.22736033052186956361457245165, −3.00358038272751015884004415147, −1.59190107048832126738474506641, −1.01790011972015345687978755584, 1.45652603449643632581843197449, 2.06039982636785823145496621152, 3.34499177551626753985751950698, 3.52125660576121523932040564436, 5.42206684866313478113511377241, 5.78686281170390371835236853063, 6.55697671707859938048438871150, 7.50782227528826818430100374147, 8.289193176363329840804814233466, 8.856042623125622770221570127009

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