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Properties

Label 2-936-104.61-c1-0-15
Degree $2$
Conductor $936$
Sign $0.995 + 0.0941i$
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  + (0.628 − 1.26i)2-s + (−1.20 − 1.59i)4-s + 2.59i·5-s + (−0.300 − 0.520i)7-s + (−2.77 + 0.530i)8-s + (3.29 + 1.63i)10-s + (2.40 + 1.39i)11-s + (3.60 + 0.0531i)13-s + (−0.848 + 0.0534i)14-s + (−1.07 + 3.85i)16-s + (1.29 + 2.24i)17-s + (−4.38 + 2.52i)19-s + (4.14 − 3.14i)20-s + (3.27 − 2.17i)22-s + (−2.94 + 5.09i)23-s + ⋯
L(s)  = 1  + (0.444 − 0.895i)2-s + (−0.604 − 0.796i)4-s + 1.16i·5-s + (−0.113 − 0.196i)7-s + (−0.982 + 0.187i)8-s + (1.04 + 0.516i)10-s + (0.726 + 0.419i)11-s + (0.999 + 0.0147i)13-s + (−0.226 + 0.0142i)14-s + (−0.268 + 0.963i)16-s + (0.313 + 0.543i)17-s + (−1.00 + 0.580i)19-s + (0.926 − 0.703i)20-s + (0.698 − 0.464i)22-s + (−0.613 + 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.995 + 0.0941i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.995 + 0.0941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81705 - 0.0857083i\)
\(L(\frac12)\) \(\approx\) \(1.81705 - 0.0857083i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.628 + 1.26i)T \)
3 \( 1 \)
13 \( 1 + (-3.60 - 0.0531i)T \)
good5 \( 1 - 2.59iT - 5T^{2} \)
7 \( 1 + (0.300 + 0.520i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.40 - 1.39i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.29 - 2.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.38 - 2.52i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.94 - 5.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.54 - 0.889i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 + (-8.82 - 5.09i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.26 + 9.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.44 + 4.87i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.13T + 47T^{2} \)
53 \( 1 - 6.01iT - 53T^{2} \)
59 \( 1 + (9.44 - 5.45i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.84 - 1.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.39 + 1.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.230 + 0.399i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 + 7.83T + 79T^{2} \)
83 \( 1 - 0.930iT - 83T^{2} \)
89 \( 1 + (1.17 - 2.04i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.91 - 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.44019163398918517029373444594, −9.470532285285436690358712544711, −8.575959490159329953636854032327, −7.43473346838308902416032351327, −6.29722435589832709199114610878, −5.90672763024424866288896945072, −4.28181032834463655515227898483, −3.71553732978424948853609024460, −2.65002970507609884959871464752, −1.42852694593554879625841520246, 0.844736280312837531619281149321, 2.84583817945500304273158544632, 4.26074524832247325627903238134, 4.62601700046206293948992628309, 6.05636605161639819349609072729, 6.23359876615362524783767091024, 7.61761530350447837441770086114, 8.477831031304357912135314619616, 8.906222417807237284751304950464, 9.689091213991153261002063095009

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