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Properties

Label 2-936-24.11-c1-0-26
Degree $2$
Conductor $936$
Sign $0.309 + 0.950i$
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  + (0.141 − 1.40i)2-s + (−1.95 − 0.398i)4-s + 0.802·5-s + 1.93i·7-s + (−0.838 + 2.70i)8-s + (0.113 − 1.12i)10-s − 1.56i·11-s + i·13-s + (2.72 + 0.274i)14-s + (3.68 + 1.56i)16-s − 3.54i·17-s + 4.68·19-s + (−1.57 − 0.320i)20-s + (−2.19 − 0.221i)22-s + 8.86·23-s + ⋯
L(s)  = 1  + (0.100 − 0.994i)2-s + (−0.979 − 0.199i)4-s + 0.359·5-s + 0.732i·7-s + (−0.296 + 0.955i)8-s + (0.0359 − 0.357i)10-s − 0.471i·11-s + 0.277i·13-s + (0.728 + 0.0733i)14-s + (0.920 + 0.390i)16-s − 0.860i·17-s + 1.07·19-s + (−0.351 − 0.0715i)20-s + (−0.468 − 0.0471i)22-s + 1.84·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31635 - 0.956078i\)
\(L(\frac12)\) \(\approx\) \(1.31635 - 0.956078i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.141 + 1.40i)T \)
3 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 - 0.802T + 5T^{2} \)
7 \( 1 - 1.93iT - 7T^{2} \)
11 \( 1 + 1.56iT - 11T^{2} \)
17 \( 1 + 3.54iT - 17T^{2} \)
19 \( 1 - 4.68T + 19T^{2} \)
23 \( 1 - 8.86T + 23T^{2} \)
29 \( 1 - 8.14T + 29T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 - 7.05iT - 37T^{2} \)
41 \( 1 - 4.31iT - 41T^{2} \)
43 \( 1 + 1.21T + 43T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 + 1.19T + 53T^{2} \)
59 \( 1 + 8.98iT - 59T^{2} \)
61 \( 1 - 2.66iT - 61T^{2} \)
67 \( 1 + 9.46T + 67T^{2} \)
71 \( 1 - 2.27T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 0.0205iT - 79T^{2} \)
83 \( 1 + 0.956iT - 83T^{2} \)
89 \( 1 + 4.61iT - 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−9.814859204203911527504539930302, −9.280810203980473889036873537272, −8.558439569186056058879832371404, −7.50442672288546875604915095674, −6.19063998383511490061494134072, −5.35645727884779357392164253172, −4.54314675385270432862444871294, −3.17899962271084160608084210962, −2.47197622671040830361150022825, −1.00761085158661794071318787107, 1.12166914639152238125651261919, 3.09460802646483101882375474083, 4.18367241417184508551750840895, 5.11103632635400491157125774176, 5.92161330481754456026139588790, 7.04267672619493276901544474535, 7.38210570440932235018604340413, 8.534837425581341964966650472319, 9.191546225966965701988895738442, 10.19548765850807035463073519544

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