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Properties

Label 2-3640-3640.1259-c0-0-0
Degree $2$
Conductor $3640$
Sign $0.533 + 0.846i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.5 − 0.866i)9-s + (0.500 + 0.866i)10-s + (−0.5 − 0.133i)11-s + (−0.965 + 0.258i)13-s + (0.866 − 0.500i)14-s + (0.500 + 0.866i)16-s + (−0.707 − 0.707i)18-s + (−0.133 − 0.5i)19-s + (0.965 − 0.258i)20-s + (−0.258 + 0.448i)22-s + (1.22 − 0.707i)23-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.5 − 0.866i)9-s + (0.500 + 0.866i)10-s + (−0.5 − 0.133i)11-s + (−0.965 + 0.258i)13-s + (0.866 − 0.500i)14-s + (0.500 + 0.866i)16-s + (−0.707 − 0.707i)18-s + (−0.133 − 0.5i)19-s + (0.965 − 0.258i)20-s + (−0.258 + 0.448i)22-s + (1.22 − 0.707i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $0.533 + 0.846i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ 0.533 + 0.846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.199147645\)
\(L(\frac12)\) \(\approx\) \(1.199147645\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.965 - 0.258i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
53 \( 1 - 1.93T + T^{2} \)
59 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.866 - 0.5i)T^{2} \)
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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

−8.816595210155296177438549842482, −7.929579634353124143955267051534, −7.17850471110473096538119503720, −6.31840391207341097187278275794, −5.37287736784232781698425491205, −4.55173319860794712634008593996, −3.99917536171970385486932131984, −2.75667228013532794198620711307, −2.49711421293860006780234930870, −0.919251748005574016047449462849, 0.944199385558522860353883764637, 2.48293699221914773558387623262, 3.89381192668678614703437472639, 4.33668706015991623395178964021, 5.24326262744131723425626002730, 5.42944873726162305999922257881, 7.02106939125742137068964343399, 7.41049159990457292201407321455, 7.86746112000090442001770587876, 8.531156029079174192327346168585

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