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Properties

Label 2-936-936.517-c1-0-108
Degree $2$
Conductor $936$
Sign $0.434 + 0.900i$
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  + (−1.32 − 0.487i)2-s + (1.69 − 0.373i)3-s + (1.52 + 1.29i)4-s + (1.73 − 3.00i)5-s + (−2.42 − 0.328i)6-s + 2.47i·7-s + (−1.39 − 2.46i)8-s + (2.72 − 1.26i)9-s + (−3.76 + 3.14i)10-s + (−1.61 + 2.79i)11-s + (3.06 + 1.61i)12-s + (−3.34 + 1.35i)13-s + (1.20 − 3.29i)14-s + (1.81 − 5.72i)15-s + (0.653 + 3.94i)16-s + (2.89 − 5.01i)17-s + ⋯
L(s)  = 1  + (−0.938 − 0.344i)2-s + (0.976 − 0.215i)3-s + (0.762 + 0.646i)4-s + (0.775 − 1.34i)5-s + (−0.990 − 0.134i)6-s + 0.936i·7-s + (−0.493 − 0.869i)8-s + (0.907 − 0.420i)9-s + (−1.19 + 0.993i)10-s + (−0.487 + 0.843i)11-s + (0.884 + 0.467i)12-s + (−0.926 + 0.375i)13-s + (0.322 − 0.879i)14-s + (0.467 − 1.47i)15-s + (0.163 + 0.986i)16-s + (0.701 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.434 + 0.900i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.434 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43133 - 0.898666i\)
\(L(\frac12)\) \(\approx\) \(1.43133 - 0.898666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.487i)T \)
3 \( 1 + (-1.69 + 0.373i)T \)
13 \( 1 + (3.34 - 1.35i)T \)
good5 \( 1 + (-1.73 + 3.00i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.47iT - 7T^{2} \)
11 \( 1 + (1.61 - 2.79i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.89 + 5.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.26 + 3.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 9.31T + 23T^{2} \)
29 \( 1 + (3.40 + 1.96i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.59 - 3.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.66 + 2.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.56iT - 41T^{2} \)
43 \( 1 + 9.40iT - 43T^{2} \)
47 \( 1 + (7.93 - 4.57i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.880iT - 53T^{2} \)
59 \( 1 + (-4.67 - 8.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 12.6iT - 61T^{2} \)
67 \( 1 + 3.65T + 67T^{2} \)
71 \( 1 + (3.31 + 1.91i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + (5.76 + 9.99i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.07 + 1.86i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.67 - 0.964i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.53iT - 97T^{2} \)
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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

−9.550286548385151835811706696527, −9.088987661161376868761418268954, −8.681586515020954770600510034464, −7.48077040965802014854332265496, −7.01040290588439635361953595146, −5.40939807609641499522197616645, −4.67209280832052748577046586245, −2.88232555875016317050562788605, −2.26920103125289776549597752972, −1.08629804717760564042892262567, 1.46478118585463904424441606733, 2.79599798944543426730691111469, 3.37957626673582434836735811855, 5.15264707265935279447055567600, 6.24885429073123646576956806240, 7.06493655858251159553227894951, 7.75691997805107698673856682838, 8.378465636182048336616705988597, 9.704329892686110152046446505937, 9.964704638453795236912296838263

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