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Properties

Label 2-936-13.9-c1-0-10
Degree $2$
Conductor $936$
Sign $0.0128 + 0.999i$
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  − 2·5-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (−1 − 3.46i)13-s + (1.5 + 2.59i)17-s + (−3.5 − 6.06i)19-s + (0.5 − 0.866i)23-s − 25-s + (1.5 − 2.59i)29-s + 8·31-s + (−1 − 1.73i)35-s + (0.5 − 0.866i)37-s + (5.5 − 9.52i)41-s + (−5.5 − 9.52i)43-s − 12·47-s + ⋯
L(s)  = 1  − 0.894·5-s + (0.188 + 0.327i)7-s + (0.150 − 0.261i)11-s + (−0.277 − 0.960i)13-s + (0.363 + 0.630i)17-s + (−0.802 − 1.39i)19-s + (0.104 − 0.180i)23-s − 0.200·25-s + (0.278 − 0.482i)29-s + 1.43·31-s + (−0.169 − 0.292i)35-s + (0.0821 − 0.142i)37-s + (0.858 − 1.48i)41-s + (−0.838 − 1.45i)43-s − 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.0128 + 0.999i$
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: \((2,\ 936,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.695561 - 0.686698i\)
\(L(\frac12)\) \(\approx\) \(0.695561 - 0.686698i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 3.46i)T \)
good5 \( 1 + 2T + 5T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.5 + 9.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.5 + 4.33i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.979051827017120126522121919113, −8.743792395601601116507894921712, −8.286180453632167103186215693781, −7.41150037677736482965828580852, −6.47207486014047733968658970430, −5.43579378947994506828995567896, −4.47350376096098879563311590765, −3.49254939702464731708262398725, −2.36156091581789677249854122678, −0.48229332649765629059210671954, 1.44528577085793828625755493035, 2.98284346578135534053916083040, 4.15658829801098371787483478604, 4.69486923137190466252670355951, 6.09546056157668902580194699258, 6.94015731950006848119829522496, 7.84731209743166424587113228350, 8.372743001313330095481593807340, 9.571098261700932637494922552183, 10.14946519399777194995904144554

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