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Properties

Label 2-936-104.61-c1-0-26
Degree $2$
Conductor $936$
Sign $0.411 - 0.911i$
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  + (1.39 − 0.252i)2-s + (1.87 − 0.702i)4-s + 4.18i·5-s + (0.818 + 1.41i)7-s + (2.42 − 1.45i)8-s + (1.05 + 5.81i)10-s + (−0.112 − 0.0649i)11-s + (−2.30 + 2.77i)13-s + (1.49 + 1.76i)14-s + (3.01 − 2.63i)16-s + (−1.60 − 2.77i)17-s + (−3.31 + 1.91i)19-s + (2.93 + 7.83i)20-s + (−0.172 − 0.0619i)22-s + (1.09 − 1.89i)23-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)2-s + (0.936 − 0.351i)4-s + 1.87i·5-s + (0.309 + 0.535i)7-s + (0.858 − 0.512i)8-s + (0.333 + 1.84i)10-s + (−0.0339 − 0.0195i)11-s + (−0.640 + 0.768i)13-s + (0.400 + 0.472i)14-s + (0.753 − 0.657i)16-s + (−0.388 − 0.672i)17-s + (−0.761 + 0.439i)19-s + (0.656 + 1.75i)20-s + (−0.0368 − 0.0132i)22-s + (0.227 − 0.394i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.411 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44726 + 1.57985i\)
\(L(\frac12)\) \(\approx\) \(2.44726 + 1.57985i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.252i)T \)
3 \( 1 \)
13 \( 1 + (2.30 - 2.77i)T \)
good5 \( 1 - 4.18iT - 5T^{2} \)
7 \( 1 + (-0.818 - 1.41i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.112 + 0.0649i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.60 + 2.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.31 - 1.91i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.09 + 1.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.98 - 2.87i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.16T + 31T^{2} \)
37 \( 1 + (0.156 + 0.0902i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.25 - 3.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.84 + 3.37i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.71T + 47T^{2} \)
53 \( 1 + 4.75iT - 53T^{2} \)
59 \( 1 + (-9.37 + 5.40i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.24 - 0.719i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.0 - 6.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.53 + 4.39i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.28T + 73T^{2} \)
79 \( 1 + 9.44T + 79T^{2} \)
83 \( 1 - 5.48iT - 83T^{2} \)
89 \( 1 + (-0.386 + 0.669i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.07 + 8.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.40715988749989923084258829070, −9.777714460968612808850470311608, −8.396773506793470274076921505071, −7.19129094974930404829576791989, −6.77456894424260866715052396901, −5.98721676042407838656046698256, −4.86795830312065607915445282471, −3.84091806903815409661487960206, −2.72031028511197739526964961144, −2.18998918951649578021351316928, 1.02705861240088552559774818618, 2.39911582239761337309953106863, 4.00625338715234938445857032184, 4.59618328397103338579248191833, 5.31543119157183941340818857442, 6.20128374247053429575422980584, 7.42408155729702438559230863212, 8.160411706399299592315118562579, 8.834716892349648884825063333902, 10.00855496443452203936785724719

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