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Properties

Label 2-936-117.94-c1-0-21
Degree $2$
Conductor $936$
Sign $0.864 + 0.501i$
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  + (−1.11 + 1.32i)3-s + (−0.816 + 1.41i)5-s − 0.664·7-s + (−0.511 − 2.95i)9-s + (2.30 − 3.98i)11-s + (−3.59 − 0.333i)13-s + (−0.963 − 2.66i)15-s + (0.691 − 1.19i)17-s + (2.59 − 4.50i)19-s + (0.741 − 0.881i)21-s − 4.31·23-s + (1.16 + 2.01i)25-s + (4.48 + 2.62i)27-s + (0.173 − 0.301i)29-s + (2.78 − 4.81i)31-s + ⋯
L(s)  = 1  + (−0.644 + 0.764i)3-s + (−0.365 + 0.632i)5-s − 0.251·7-s + (−0.170 − 0.985i)9-s + (0.693 − 1.20i)11-s + (−0.995 − 0.0923i)13-s + (−0.248 − 0.686i)15-s + (0.167 − 0.290i)17-s + (0.596 − 1.03i)19-s + (0.161 − 0.192i)21-s − 0.899·23-s + (0.233 + 0.403i)25-s + (0.863 + 0.504i)27-s + (0.0323 − 0.0559i)29-s + (0.499 − 0.865i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.864 + 0.501i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.864 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866416 - 0.233117i\)
\(L(\frac12)\) \(\approx\) \(0.866416 - 0.233117i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 - 1.32i)T \)
13 \( 1 + (3.59 + 0.333i)T \)
good5 \( 1 + (0.816 - 1.41i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.664T + 7T^{2} \)
11 \( 1 + (-2.30 + 3.98i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.691 + 1.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.59 + 4.50i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.31T + 23T^{2} \)
29 \( 1 + (-0.173 + 0.301i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.78 + 4.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.78 - 4.82i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.66T + 41T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 + (5.07 + 8.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.833T + 53T^{2} \)
59 \( 1 + (3.54 + 6.13i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 6.99T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + (2.13 - 3.69i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.70T + 73T^{2} \)
79 \( 1 + (7.71 + 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.471 - 0.817i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.06 + 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−9.878419180791618571442935895623, −9.472246559568779681050456034238, −8.402195770326616996209214939207, −7.29797368170059121376702911667, −6.48216230386278604729915622831, −5.66775179562381048528943192105, −4.66442103371032372689785229919, −3.64174039381168012398263989782, −2.81590950528429591892257548239, −0.53115512715778705967175980273, 1.17999876900789682734549480311, 2.37695416221012475140482471975, 4.07462743221885881080675320182, 4.86849725021575596036191480494, 5.87012376630107035647777209883, 6.76764044738724632052273901092, 7.58189047654200208880840118623, 8.198700625664389115865916360818, 9.445691342945776111020921011405, 10.03012316467969760890347725523

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