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Properties

Label 2-3640-3640.1357-c0-0-3
Degree $2$
Conductor $3640$
Sign $0.749 + 0.661i$
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  − 2-s + (1 − i)3-s + 4-s + 5-s + (−1 + i)6-s + i·7-s − 8-s − i·9-s − 10-s + (1 − i)12-s − i·13-s − i·14-s + (1 − i)15-s + 16-s + i·18-s + (1 + i)19-s + ⋯
L(s)  = 1  − 2-s + (1 − i)3-s + 4-s + 5-s + (−1 + i)6-s + i·7-s − 8-s − i·9-s − 10-s + (1 − i)12-s − i·13-s − i·14-s + (1 − i)15-s + 16-s + i·18-s + (1 + i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ 0.749 + 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.450478694\)
\(L(\frac12)\) \(\approx\) \(1.450478694\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - iT \)
13 \( 1 + iT \)
good3 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1 + i)T - iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1 - i)T - iT^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 + iT^{2} \)
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

−8.526097705652569631043125398098, −8.088109188591817011528067979435, −7.42612635641757739097356001697, −6.52601926284271183870333854701, −5.93104534932442058382987618492, −5.25164847792684728346462356742, −3.40299070731961812233422288517, −2.63656746494652863087634338946, −2.09077439571503648448042010353, −1.22171608947971996866647050315, 1.32430311189222599238350396016, 2.31638900318484713923164257512, 3.18385550506375721832231489964, 3.99557380282224515954930302796, 4.94264243918414326049561171181, 5.95822718955722050245622082821, 6.85791608019722800216962287479, 7.44107739507239119673771621843, 8.287923391889168295214624734964, 9.147636984746517294532014615999

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