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Properties

Label 2-936-8.5-c1-0-13
Degree $2$
Conductor $936$
Sign $-0.0864 - 0.996i$
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  + (−0.671 + 1.24i)2-s + (−1.09 − 1.67i)4-s + i·5-s − 0.146·7-s + (2.81 − 0.244i)8-s + (−1.24 − 0.671i)10-s − 2.68i·11-s + i·13-s + (0.0982 − 0.182i)14-s + (−1.58 + 3.67i)16-s − 17-s + 4i·19-s + (1.67 − 1.09i)20-s + (3.34 + 1.80i)22-s + 6.68·23-s + ⋯
L(s)  = 1  + (−0.474 + 0.880i)2-s + (−0.549 − 0.835i)4-s + 0.447i·5-s − 0.0553·7-s + (0.996 − 0.0864i)8-s + (−0.393 − 0.212i)10-s − 0.809i·11-s + 0.277i·13-s + (0.0262 − 0.0486i)14-s + (−0.396 + 0.917i)16-s − 0.242·17-s + 0.917i·19-s + (0.373 − 0.245i)20-s + (0.712 + 0.384i)22-s + 1.39·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0864 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.0864 - 0.996i$
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: \((2,\ 936,\ (\ :1/2),\ -0.0864 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.751393 + 0.819455i\)
\(L(\frac12)\) \(\approx\) \(0.751393 + 0.819455i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.671 - 1.24i)T \)
3 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 - iT - 5T^{2} \)
7 \( 1 + 0.146T + 7T^{2} \)
11 \( 1 + 2.68iT - 11T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 6.68T + 23T^{2} \)
29 \( 1 - 4.39iT - 29T^{2} \)
31 \( 1 - 1.31T + 31T^{2} \)
37 \( 1 - 3.97iT - 37T^{2} \)
41 \( 1 - 6.39T + 41T^{2} \)
43 \( 1 + 6.83iT - 43T^{2} \)
47 \( 1 - 7.12T + 47T^{2} \)
53 \( 1 - 8.97iT - 53T^{2} \)
59 \( 1 - 12.3iT - 59T^{2} \)
61 \( 1 - 8.35iT - 61T^{2} \)
67 \( 1 - 8.29iT - 67T^{2} \)
71 \( 1 + 5.51T + 71T^{2} \)
73 \( 1 + 6.97T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 4.29iT - 83T^{2} \)
89 \( 1 - 5.37T + 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−10.35821834959610817754541039131, −9.090427093900938229636935988472, −8.745073024929090662986261242694, −7.64614281651411384894698906936, −6.94248795277497247513177566595, −6.12555304966974498309804541095, −5.29950420321947697786159665165, −4.19095662217913230636622011933, −2.90392742272844534258131750993, −1.16979071573393543096671472218, 0.75358876853674065754331025359, 2.16841775700042522476603607332, 3.21191133713263573914301049545, 4.48452113151079999515526324801, 5.06362553605733191639499257715, 6.61811264225515818864958454006, 7.50995183480830846537455720500, 8.364116240287790032442154611310, 9.260649129257403307336542441299, 9.665526674803454606994950337115

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