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Properties

Label 2-3640-3640.1139-c0-0-0
Degree $2$
Conductor $3640$
Sign $0.912 + 0.408i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.258 − 0.965i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.5 − 0.866i)9-s + (0.499 + 0.866i)10-s + (1.86 + 0.5i)11-s + (0.258 − 0.965i)13-s + (−0.866 − 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)18-s + (−0.133 − 0.5i)19-s + (−0.707 − 0.707i)20-s − 1.93·22-s + (1.67 + 0.965i)23-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.258 − 0.965i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.5 − 0.866i)9-s + (0.499 + 0.866i)10-s + (1.86 + 0.5i)11-s + (0.258 − 0.965i)13-s + (−0.866 − 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)18-s + (−0.133 − 0.5i)19-s + (−0.707 − 0.707i)20-s − 1.93·22-s + (1.67 + 0.965i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $0.912 + 0.408i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ 0.912 + 0.408i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.059294469\)
\(L(\frac12)\) \(\approx\) \(1.059294469\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-0.258 + 0.965i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.859777488211715415707509645969, −8.088716470381630616077252869074, −7.27496311700854080411914126390, −6.63343308870734315892395417121, −5.76194406093971091565208011883, −5.03935071196643924389188025537, −4.10311967670417955624743730774, −3.06067256349742275380775679976, −1.56182293132821475819546228291, −1.11630158868137036278677730475, 1.27784901614642158122278792216, 1.97067431621294319459787672693, 3.25702144696155195004480855806, 3.94036019106078573145704120808, 4.76826887214581028525332840486, 6.28483459564821395998708386722, 6.85093488868126614452290521132, 7.16586295535002886416043724807, 8.216507154056533943953088652757, 8.608723645902910743691499614337

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