Properties

Label 2-936-13.9-c1-0-11
Degree $2$
Conductor $936$
Sign $0.657 + 0.753i$
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  + 1.85·5-s + (−1.42 − 2.47i)7-s + (0.649 − 1.12i)11-s + (3.00 + 1.99i)13-s + (0.278 + 0.481i)17-s + (−0.649 − 1.12i)19-s + (4.50 − 7.80i)23-s − 1.55·25-s + (2.22 − 3.85i)29-s − 0.443·31-s + (−2.64 − 4.58i)35-s + (−3.78 + 6.55i)37-s + (5.43 − 9.41i)41-s + (−0.871 − 1.50i)43-s + 5.01·47-s + ⋯
L(s)  = 1  + 0.829·5-s + (−0.539 − 0.934i)7-s + (0.195 − 0.339i)11-s + (0.833 + 0.552i)13-s + (0.0674 + 0.116i)17-s + (−0.149 − 0.258i)19-s + (0.939 − 1.62i)23-s − 0.311·25-s + (0.413 − 0.716i)29-s − 0.0797·31-s + (−0.447 − 0.775i)35-s + (−0.622 + 1.07i)37-s + (0.848 − 1.46i)41-s + (−0.132 − 0.230i)43-s + 0.730·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.657 + 0.753i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59472 - 0.725021i\)
\(L(\frac12)\) \(\approx\) \(1.59472 - 0.725021i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (-3.00 - 1.99i)T \)
good5 \( 1 - 1.85T + 5T^{2} \)
7 \( 1 + (1.42 + 2.47i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.649 + 1.12i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.278 - 0.481i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.649 + 1.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.50 + 7.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.22 + 3.85i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.443T + 31T^{2} \)
37 \( 1 + (3.78 - 6.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.43 + 9.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.871 + 1.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.01T + 47T^{2} \)
53 \( 1 + 2.14T + 53T^{2} \)
59 \( 1 + (-4.41 - 7.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.86 - 8.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.43 + 9.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.505 + 0.875i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.112T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 4.72T + 83T^{2} \)
89 \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.07 - 1.86i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.10275632167449720765414484560, −9.077671009776925768747582769447, −8.488384495764227734508476744061, −7.18881280881636020032227837844, −6.52001735300601824393222471479, −5.79377152817459760883607782082, −4.52533468937468644142254912917, −3.63534838780565046081090561581, −2.36859558311180090167448744886, −0.910009235901005887578509586768, 1.48207794220660148924304676577, 2.71025441677068183972321901993, 3.70665510009137394452494833456, 5.21554677135614228170074881368, 5.79649132682586187550852956466, 6.59140137502682125387652438222, 7.65328409542865594573906640787, 8.723139860877652675583681692183, 9.371559535432763115991080243167, 9.977559595482804604029113665675

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