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Properties

Label 2-936-117.16-c1-0-31
Degree $2$
Conductor $936$
Sign $0.399 + 0.916i$
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  + (1.17 − 1.27i)3-s + (−0.185 − 0.320i)5-s + (−0.110 − 0.191i)7-s + (−0.254 − 2.98i)9-s + 4.20·11-s + (3.45 + 1.03i)13-s + (−0.625 − 0.139i)15-s + (−1.24 + 2.16i)17-s + (0.226 − 0.391i)19-s + (−0.373 − 0.0832i)21-s + (1.02 − 1.77i)23-s + (2.43 − 4.21i)25-s + (−4.11 − 3.17i)27-s − 3.70·29-s + (−0.816 − 1.41i)31-s + ⋯
L(s)  = 1  + (0.676 − 0.736i)3-s + (−0.0827 − 0.143i)5-s + (−0.0417 − 0.0722i)7-s + (−0.0846 − 0.996i)9-s + 1.26·11-s + (0.957 + 0.287i)13-s + (−0.161 − 0.0360i)15-s + (−0.302 + 0.524i)17-s + (0.0518 − 0.0898i)19-s + (−0.0814 − 0.0181i)21-s + (0.213 − 0.370i)23-s + (0.486 − 0.842i)25-s + (−0.791 − 0.611i)27-s − 0.687·29-s + (−0.146 − 0.253i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.399 + 0.916i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.399 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76053 - 1.15383i\)
\(L(\frac12)\) \(\approx\) \(1.76053 - 1.15383i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 + 1.27i)T \)
13 \( 1 + (-3.45 - 1.03i)T \)
good5 \( 1 + (0.185 + 0.320i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.110 + 0.191i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.20T + 11T^{2} \)
17 \( 1 + (1.24 - 2.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.226 + 0.391i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.02 + 1.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.70T + 29T^{2} \)
31 \( 1 + (0.816 + 1.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.00 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.03 + 3.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.75 - 9.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.918 + 1.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.12T + 53T^{2} \)
59 \( 1 + 4.81T + 59T^{2} \)
61 \( 1 + (-3.91 - 6.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.82 + 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.89 - 11.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.58T + 73T^{2} \)
79 \( 1 + (6.36 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.54 - 9.60i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.114 + 0.197i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.11 + 8.85i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.626451160943720938685135110563, −8.890755334994777829649542890695, −8.411079873636622060340953873688, −7.31955563365344731535300486093, −6.57888178843916349768479763382, −5.86444010022863667060263381588, −4.24235357415426178707734565943, −3.58094025231302669359829814072, −2.19936289252534201138857521833, −1.05691715273599415156514653649, 1.56874827710208793578094507650, 3.08926506083711353218836535772, 3.77160750742354677300781144225, 4.77579479270366890059814129167, 5.83676980837081976331168835922, 6.89387043437909856734505862033, 7.79112715268083366468092537757, 8.903266696131752477230113144230, 9.126863789842448068080729835497, 10.13582463843776659634948854853

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