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Properties

Label 2-964-964.83-c0-0-0
Degree $2$
Conductor $964$
Sign $0.358 - 0.933i$
Analytic cond. $0.481098$
Root an. cond. $0.693612$
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 + (0.198 + 0.0646i)5-s + 0.999i·8-s + (0.104 + 0.994i)9-s + (0.139 + 0.155i)10-s + (−0.715 + 0.0375i)13-s + (−0.5 + 0.866i)16-s + (−0.292 − 1.84i)17-s + (−0.406 + 0.913i)18-s + (0.0434 + 0.204i)20-s + (−0.773 − 0.562i)25-s + (−0.638 − 0.325i)26-s + (1.89 − 0.198i)29-s + (−0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.198 + 0.0646i)5-s + 0.999i·8-s + (0.104 + 0.994i)9-s + (0.139 + 0.155i)10-s + (−0.715 + 0.0375i)13-s + (−0.5 + 0.866i)16-s + (−0.292 − 1.84i)17-s + (−0.406 + 0.913i)18-s + (0.0434 + 0.204i)20-s + (−0.773 − 0.562i)25-s + (−0.638 − 0.325i)26-s + (1.89 − 0.198i)29-s + (−0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(964\)    =    \(2^{2} \cdot 241\)
Sign: $0.358 - 0.933i$
Analytic conductor: \(0.481098\)
Root analytic conductor: \(0.693612\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{964} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 964,\ (\ :0),\ 0.358 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.642418981\)
\(L(\frac12)\) \(\approx\) \(1.642418981\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
241 \( 1 + T \)
good3 \( 1 + (-0.104 - 0.994i)T^{2} \)
5 \( 1 + (-0.198 - 0.0646i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (0.743 + 0.669i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.715 - 0.0375i)T + (0.994 - 0.104i)T^{2} \)
17 \( 1 + (0.292 + 1.84i)T + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (-1.89 + 0.198i)T + (0.978 - 0.207i)T^{2} \)
31 \( 1 + (-0.743 - 0.669i)T^{2} \)
37 \( 1 + (-1.41 - 0.0740i)T + (0.994 + 0.104i)T^{2} \)
41 \( 1 + (1.01 - 1.40i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.951 - 0.309i)T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.809 + 1.81i)T + (-0.669 - 0.743i)T^{2} \)
59 \( 1 + (-0.104 + 0.994i)T^{2} \)
61 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.669 - 0.743i)T^{2} \)
71 \( 1 + (0.207 + 0.978i)T^{2} \)
73 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.104 - 0.994i)T^{2} \)
89 \( 1 + (-0.101 + 0.0270i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \)
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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

−10.38044479638583364120030573736, −9.628053112832149151695300595920, −8.422390291105223262466549002653, −7.70948496725832684139642895818, −6.93817196092851800154570433586, −6.08717787624250185377669598718, −4.86469587879325740051325354407, −4.66509109077911241430454429329, −3.03900137196083437728071566348, −2.24876865051149137191498878156, 1.45269497930259395368550556650, 2.73995022396591210710021843790, 3.82702301932983620381491729308, 4.58518250105455066379376841583, 5.80154969626044393797574051687, 6.34648508047860459324323878930, 7.30612370373266591443153960200, 8.524151426733974619951951495037, 9.486433131903012723774276073443, 10.19723662096975880037576429388

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