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Properties

Label 2-936-9.7-c1-0-25
Degree 22
Conductor 936936
Sign −0.266+0.963i-0.266 + 0.963i
Analytic cond. 7.473997.47399
Root an. cond. 2.733862.73386
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.793 − 1.53i)3-s + (0.671 + 1.16i)5-s + (2.29 − 3.98i)7-s + (−1.74 + 2.44i)9-s + (0.873 − 1.51i)11-s + (−0.5 − 0.866i)13-s + (1.25 − 1.95i)15-s + 0.936·17-s + 1.28·19-s + (−7.95 − 0.378i)21-s + (0.711 + 1.23i)23-s + (1.59 − 2.76i)25-s + (5.14 + 0.739i)27-s + (−1.89 + 3.28i)29-s + (−3.71 − 6.43i)31-s + ⋯
L(s)  = 1  + (−0.458 − 0.888i)3-s + (0.300 + 0.520i)5-s + (0.869 − 1.50i)7-s + (−0.580 + 0.814i)9-s + (0.263 − 0.456i)11-s + (−0.138 − 0.240i)13-s + (0.324 − 0.505i)15-s + 0.227·17-s + 0.295·19-s + (−1.73 − 0.0826i)21-s + (0.148 + 0.256i)23-s + (0.319 − 0.553i)25-s + (0.989 + 0.142i)27-s + (−0.352 + 0.610i)29-s + (−0.667 − 1.15i)31-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((−0.266+0.963i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((−0.266+0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: −0.266+0.963i-0.266 + 0.963i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(313,⋅)\chi_{936} (313, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), −0.266+0.963i)(2,\ 936,\ (\ :1/2),\ -0.266 + 0.963i)

Particular Values

L(1)L(1) ≈\approx 0.869330−1.14222i0.869330 - 1.14222i
L(12)L(\frac12) ≈\approx 0.869330−1.14222i0.869330 - 1.14222i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(0.793+1.53i)T 1 + (0.793 + 1.53i)T
13 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good5 1+(−0.671−1.16i)T+(−2.5+4.33i)T2 1 + (-0.671 - 1.16i)T + (-2.5 + 4.33i)T^{2}
7 1+(−2.29+3.98i)T+(−3.5−6.06i)T2 1 + (-2.29 + 3.98i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.873+1.51i)T+(−5.5−9.52i)T2 1 + (-0.873 + 1.51i)T + (-5.5 - 9.52i)T^{2}
17 1−0.936T+17T2 1 - 0.936T + 17T^{2}
19 1−1.28T+19T2 1 - 1.28T + 19T^{2}
23 1+(−0.711−1.23i)T+(−11.5+19.9i)T2 1 + (-0.711 - 1.23i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.89−3.28i)T+(−14.5−25.1i)T2 1 + (1.89 - 3.28i)T + (-14.5 - 25.1i)T^{2}
31 1+(3.71+6.43i)T+(−15.5+26.8i)T2 1 + (3.71 + 6.43i)T + (-15.5 + 26.8i)T^{2}
37 1−3.81T+37T2 1 - 3.81T + 37T^{2}
41 1+(4.63+8.02i)T+(−20.5+35.5i)T2 1 + (4.63 + 8.02i)T + (-20.5 + 35.5i)T^{2}
43 1+(1.09−1.89i)T+(−21.5−37.2i)T2 1 + (1.09 - 1.89i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.13−1.96i)T+(−23.5−40.7i)T2 1 + (1.13 - 1.96i)T + (-23.5 - 40.7i)T^{2}
53 1−3.11T+53T2 1 - 3.11T + 53T^{2}
59 1+(−0.361−0.626i)T+(−29.5+51.0i)T2 1 + (-0.361 - 0.626i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.607+1.05i)T+(−30.5−52.8i)T2 1 + (-0.607 + 1.05i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.54+4.40i)T+(−33.5+58.0i)T2 1 + (2.54 + 4.40i)T + (-33.5 + 58.0i)T^{2}
71 1+4.34T+71T2 1 + 4.34T + 71T^{2}
73 1+15.9T+73T2 1 + 15.9T + 73T^{2}
79 1+(−4.70+8.15i)T+(−39.5−68.4i)T2 1 + (-4.70 + 8.15i)T + (-39.5 - 68.4i)T^{2}
83 1+(−5.04+8.74i)T+(−41.5−71.8i)T2 1 + (-5.04 + 8.74i)T + (-41.5 - 71.8i)T^{2}
89 1−16.6T+89T2 1 - 16.6T + 89T^{2}
97 1+(4.56−7.90i)T+(−48.5−84.0i)T2 1 + (4.56 - 7.90i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.13886214156580131051411916565, −8.837486954964441517504388064387, −7.75022050713866490950328413979, −7.38166133023084474158441925877, −6.50390644520274978060753912758, −5.58866053658057178717224899887, −4.56446253269558715513320656143, −3.35259962327964497239861808323, −1.91854730772275985281308253331, −0.75813660728091640594898891803, 1.61178769264283595394519327053, 2.94032196507401585094190844429, 4.33422600612281086517198280007, 5.17638242265693078434738438899, 5.60506027791632978653446951859, 6.70527781620013711256388974865, 8.051622403352503330430740677507, 8.928708904448290596754249316349, 9.314498265239257353094625307415, 10.21842068630486917504941353522

Graph of the ZZ-function along the critical line