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Properties

Label 2-936-9.7-c1-0-23
Degree $2$
Conductor $936$
Sign $0.562 + 0.826i$
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  + (−1.06 + 1.36i)3-s + (0.662 + 1.14i)5-s + (0.798 − 1.38i)7-s + (−0.736 − 2.90i)9-s + (2.97 − 5.14i)11-s + (−0.5 − 0.866i)13-s + (−2.27 − 0.315i)15-s − 4.63·17-s − 5.56·19-s + (1.04 + 2.56i)21-s + (−2.13 − 3.69i)23-s + (1.62 − 2.81i)25-s + (4.75 + 2.08i)27-s + (0.256 − 0.444i)29-s + (−4.53 − 7.85i)31-s + ⋯
L(s)  = 1  + (−0.614 + 0.789i)3-s + (0.296 + 0.513i)5-s + (0.301 − 0.522i)7-s + (−0.245 − 0.969i)9-s + (0.896 − 1.55i)11-s + (−0.138 − 0.240i)13-s + (−0.586 − 0.0813i)15-s − 1.12·17-s − 1.27·19-s + (0.227 + 0.559i)21-s + (−0.444 − 0.769i)23-s + (0.324 − 0.562i)25-s + (0.915 + 0.401i)27-s + (0.0476 − 0.0825i)29-s + (−0.814 − 1.41i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.562 + 0.826i$
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: \((2,\ 936,\ (\ :1/2),\ 0.562 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.932368 - 0.493508i\)
\(L(\frac12)\) \(\approx\) \(0.932368 - 0.493508i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.06 - 1.36i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.662 - 1.14i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.798 + 1.38i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.97 + 5.14i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.63T + 17T^{2} \)
19 \( 1 + 5.56T + 19T^{2} \)
23 \( 1 + (2.13 + 3.69i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.256 + 0.444i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.53 + 7.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.53T + 37T^{2} \)
41 \( 1 + (-3.29 - 5.71i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.10 + 5.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.15 + 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 + (-6.13 - 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.99 - 6.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.29 + 9.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.26T + 71T^{2} \)
73 \( 1 + 7.18T + 73T^{2} \)
79 \( 1 + (6.24 - 10.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.20 + 14.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.38T + 89T^{2} \)
97 \( 1 + (0.100 - 0.174i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.34883550302207327936407034501, −8.941841401608539207015325751033, −8.651813437225516842122307846148, −7.18170107730609457306295863034, −6.24665830500748801070129272797, −5.81645976733113332036178776944, −4.37583195455343830670492586958, −3.87452791774077024717427744890, −2.48992616786068114104644301471, −0.53756610497582452961131555863, 1.56740930325394201464042244166, 2.22079824113955840580666246497, 4.21593382905361811350583984671, 4.97893854296286148740281215349, 5.91973378346278822145357778110, 6.85379143241313817253998580958, 7.39694252364593955416571233485, 8.706324831292124916987304200091, 9.121968655390746568119361334476, 10.26779897097082981295249085458

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