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Properties

Label 2-936-8.5-c1-0-31
Degree $2$
Conductor $936$
Sign $0.287 - 0.957i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 + 0.137i)2-s + (1.96 + 0.386i)4-s + 4.18i·5-s + 2.70·7-s + (2.70 + 0.814i)8-s + (−0.575 + 5.89i)10-s + 0.956i·11-s − i·13-s + (3.81 + 0.372i)14-s + (3.70 + 1.51i)16-s − 4.96·17-s − 5.59i·19-s + (−1.61 + 8.21i)20-s + (−0.131 + 1.34i)22-s − 5.16·23-s + ⋯
L(s)  = 1  + (0.995 + 0.0971i)2-s + (0.981 + 0.193i)4-s + 1.87i·5-s + 1.02·7-s + (0.957 + 0.287i)8-s + (−0.181 + 1.86i)10-s + 0.288i·11-s − 0.277i·13-s + (1.01 + 0.0995i)14-s + (0.925 + 0.379i)16-s − 1.20·17-s − 1.28i·19-s + (−0.362 + 1.83i)20-s + (−0.0280 + 0.287i)22-s − 1.07·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.287 - 0.957i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60594 + 1.93774i\)
\(L(\frac12)\) \(\approx\) \(2.60594 + 1.93774i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.137i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 - 4.18iT - 5T^{2} \)
7 \( 1 - 2.70T + 7T^{2} \)
11 \( 1 - 0.956iT - 11T^{2} \)
17 \( 1 + 4.96T + 17T^{2} \)
19 \( 1 + 5.59iT - 19T^{2} \)
23 \( 1 + 5.16T + 23T^{2} \)
29 \( 1 + 4.46iT - 29T^{2} \)
31 \( 1 - 6.37T + 31T^{2} \)
37 \( 1 - 1.66iT - 37T^{2} \)
41 \( 1 - 8.64T + 41T^{2} \)
43 \( 1 - 7.67iT - 43T^{2} \)
47 \( 1 - 8.01T + 47T^{2} \)
53 \( 1 - 0.0797iT - 53T^{2} \)
59 \( 1 + 4.70iT - 59T^{2} \)
61 \( 1 + 8.10iT - 61T^{2} \)
67 \( 1 + 13.7iT - 67T^{2} \)
71 \( 1 + 1.15T + 71T^{2} \)
73 \( 1 - 1.66T + 73T^{2} \)
79 \( 1 - 4.30T + 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 6.17T + 97T^{2} \)
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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

−10.61357252079546777120392372386, −9.672566995577301326787913874112, −8.121169416983992387093560257303, −7.51704325340405789300701867401, −6.62723220805288959974922433717, −6.11498430116825041758617239316, −4.81142487688499007804000899193, −4.02488875027423657542696208318, −2.76280648317380913119239196641, −2.17024384776544917144359021576, 1.22900047329686835119429146226, 2.16845086415072018926118118129, 4.07060584463657153450938701487, 4.43498023885763358306155600251, 5.40656215479090694203168726377, 6.00624417171268262779197317473, 7.40201168804568835033846028453, 8.280598113993218544010484595855, 8.831189136584579001408103794154, 9.989675519606086317242759858569

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