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Properties

Label 2-936-9.7-c1-0-15
Degree $2$
Conductor $936$
Sign $0.995 - 0.0998i$
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  + (−0.215 + 1.71i)3-s + (−1.80 − 3.11i)5-s + (−2.42 + 4.20i)7-s + (−2.90 − 0.739i)9-s + (2.96 − 5.14i)11-s + (−0.5 − 0.866i)13-s + (5.74 − 2.42i)15-s + 3.14·17-s + 4.53·19-s + (−6.69 − 5.07i)21-s + (2.40 + 4.16i)23-s + (−3.98 + 6.89i)25-s + (1.89 − 4.83i)27-s + (0.967 − 1.67i)29-s + (2.30 + 4.00i)31-s + ⋯
L(s)  = 1  + (−0.124 + 0.992i)3-s + (−0.805 − 1.39i)5-s + (−0.916 + 1.58i)7-s + (−0.969 − 0.246i)9-s + (0.895 − 1.55i)11-s + (−0.138 − 0.240i)13-s + (1.48 − 0.625i)15-s + 0.762·17-s + 1.04·19-s + (−1.46 − 1.10i)21-s + (0.501 + 0.868i)23-s + (−0.796 + 1.37i)25-s + (0.364 − 0.931i)27-s + (0.179 − 0.311i)29-s + (0.414 + 0.718i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.995 - 0.0998i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.995 - 0.0998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14078 + 0.0571043i\)
\(L(\frac12)\) \(\approx\) \(1.14078 + 0.0571043i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.215 - 1.71i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.80 + 3.11i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.42 - 4.20i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.96 + 5.14i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 3.14T + 17T^{2} \)
19 \( 1 - 4.53T + 19T^{2} \)
23 \( 1 + (-2.40 - 4.16i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.967 + 1.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.30 - 4.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + (-0.389 - 0.674i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.164 - 0.285i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.01 + 3.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.03T + 53T^{2} \)
59 \( 1 + (1.75 + 3.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.77 + 4.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.91 + 5.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.18T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + (-1.97 + 3.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.28 - 2.22i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + (-2.58 + 4.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.552589428760266218726090456522, −9.380036211518316987306039489488, −8.581684473237718244374397857854, −7.999326417452499140435456059145, −6.24230607326108647081929580252, −5.58061580024683342998878095458, −4.92465323648073206103644474656, −3.61660964090936577877976566197, −3.07972334959876327356157147858, −0.77267566003477173609734947857, 0.959652382538705131504090521676, 2.65533536560560297961408318704, 3.59115615124550858199231266660, 4.49005239000959131344022350177, 6.28573873374418878925897564430, 6.81827501286846439311084844312, 7.40541242682766354586536387326, 7.76957541303410393727857446852, 9.449246094481393199035747275068, 10.10554416545879290967845117354

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