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Properties

Label 2-936-936.779-c1-0-27
Degree $2$
Conductor $936$
Sign $-0.676 - 0.736i$
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  + (−1.22 + 0.707i)2-s + (−1.68 − 0.410i)3-s + (0.999 − 1.73i)4-s + (2.22 + 1.28i)5-s + (2.35 − 0.687i)6-s + (2.15 + 3.72i)7-s + 2.82i·8-s + (2.66 + 1.38i)9-s − 3.63·10-s + (−2.39 + 2.50i)12-s + (−1.80 + 3.12i)13-s + (−5.27 − 3.04i)14-s + (−3.22 − 3.07i)15-s + (−2.00 − 3.46i)16-s + 8.24i·17-s + (−4.23 + 0.192i)18-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (−0.971 − 0.236i)3-s + (0.499 − 0.866i)4-s + (0.996 + 0.575i)5-s + (0.959 − 0.280i)6-s + (0.813 + 1.40i)7-s + 0.999i·8-s + (0.887 + 0.460i)9-s − 1.15·10-s + (−0.690 + 0.722i)12-s + (−0.499 + 0.866i)13-s + (−1.40 − 0.813i)14-s + (−0.831 − 0.794i)15-s + (−0.500 − 0.866i)16-s + 1.99i·17-s + (−0.998 + 0.0454i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.676 - 0.736i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.676 - 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350786 + 0.799166i\)
\(L(\frac12)\) \(\approx\) \(0.350786 + 0.799166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 + (1.68 + 0.410i)T \)
13 \( 1 + (1.80 - 3.12i)T \)
good5 \( 1 + (-2.22 - 1.28i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.15 - 3.72i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 8.24iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.47 + 9.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.33T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.25 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.4 - 6.01i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.57iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 - 84.0i)T^{2} \)
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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

−10.19218619982614177243594868594, −9.686543499264586486794784733346, −8.631263936576447006176599935203, −7.920320387956945847194787098500, −6.75129903887760840673633923547, −6.07400121576978734923774204899, −5.65500160554109134111098943535, −4.59964210120065697580406024754, −2.22399879421882653053189941814, −1.74446473863939941466554965097, 0.64281772866658147314111226889, 1.52693090587260880505325014533, 3.15948103848787726228957120375, 4.67621301428203937861711996372, 5.10280908689044605853252163517, 6.53160958573417204895960621467, 7.26499818377853229855139635324, 8.074880952315145204555995880331, 9.261375897842867641863650413773, 10.00333979629885138836382588732

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