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Properties

Label 2-936-13.9-c1-0-1
Degree 22
Conductor 936936
Sign −0.406−0.913i-0.406 - 0.913i
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  − 0.678·5-s + (−0.160 − 0.278i)7-s + (−2.10 + 3.65i)11-s + (−2.28 − 2.78i)13-s + (1.76 + 3.06i)17-s + (2.10 + 3.65i)19-s + (−0.787 + 1.36i)23-s − 4.53·25-s + (−4.55 + 7.89i)29-s + 2.53·31-s + (0.109 + 0.188i)35-s + (0.0175 − 0.0303i)37-s + (−1.12 + 1.95i)41-s + (3.37 + 5.85i)43-s − 5.57·47-s + ⋯
L(s)  = 1  − 0.303·5-s + (−0.0607 − 0.105i)7-s + (−0.635 + 1.10i)11-s + (−0.634 − 0.772i)13-s + (0.429 + 0.743i)17-s + (0.483 + 0.838i)19-s + (−0.164 + 0.284i)23-s − 0.907·25-s + (−0.846 + 1.46i)29-s + 0.456·31-s + (0.0184 + 0.0319i)35-s + (0.00288 − 0.00499i)37-s + (−0.175 + 0.304i)41-s + (0.515 + 0.892i)43-s − 0.813·47-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((−0.406−0.913i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((−0.406−0.913i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: −0.406−0.913i-0.406 - 0.913i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(217,⋅)\chi_{936} (217, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), −0.406−0.913i)(2,\ 936,\ (\ :1/2),\ -0.406 - 0.913i)

Particular Values

L(1)L(1) ≈\approx 0.453273+0.698114i0.453273 + 0.698114i
L(12)L(\frac12) ≈\approx 0.453273+0.698114i0.453273 + 0.698114i
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 1
13 1+(2.28+2.78i)T 1 + (2.28 + 2.78i)T
good5 1+0.678T+5T2 1 + 0.678T + 5T^{2}
7 1+(0.160+0.278i)T+(−3.5+6.06i)T2 1 + (0.160 + 0.278i)T + (-3.5 + 6.06i)T^{2}
11 1+(2.10−3.65i)T+(−5.5−9.52i)T2 1 + (2.10 - 3.65i)T + (-5.5 - 9.52i)T^{2}
17 1+(−1.76−3.06i)T+(−8.5+14.7i)T2 1 + (-1.76 - 3.06i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.10−3.65i)T+(−9.5+16.4i)T2 1 + (-2.10 - 3.65i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.787−1.36i)T+(−11.5−19.9i)T2 1 + (0.787 - 1.36i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.55−7.89i)T+(−14.5−25.1i)T2 1 + (4.55 - 7.89i)T + (-14.5 - 25.1i)T^{2}
31 1−2.53T+31T2 1 - 2.53T + 31T^{2}
37 1+(−0.0175+0.0303i)T+(−18.5−32.0i)T2 1 + (-0.0175 + 0.0303i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.12−1.95i)T+(−20.5−35.5i)T2 1 + (1.12 - 1.95i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.37−5.85i)T+(−21.5+37.2i)T2 1 + (-3.37 - 5.85i)T + (-21.5 + 37.2i)T^{2}
47 1+5.57T+47T2 1 + 5.57T + 47T^{2}
53 1+4.67T+53T2 1 + 4.67T + 53T^{2}
59 1+(−4.86−8.42i)T+(−29.5+51.0i)T2 1 + (-4.86 - 8.42i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.96+5.13i)T+(−30.5+52.8i)T2 1 + (2.96 + 5.13i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.41−11.1i)T+(−33.5−58.0i)T2 1 + (6.41 - 11.1i)T + (-33.5 - 58.0i)T^{2}
71 1+(−4.78−8.29i)T+(−35.5+61.4i)T2 1 + (-4.78 - 8.29i)T + (-35.5 + 61.4i)T^{2}
73 1+6.07T+73T2 1 + 6.07T + 73T^{2}
79 1−4.61T+79T2 1 - 4.61T + 79T^{2}
83 1+10.9T+83T2 1 + 10.9T + 83T^{2}
89 1+(2−3.46i)T+(−44.5−77.0i)T2 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2}
97 1+(2.94+5.10i)T+(−48.5+84.0i)T2 1 + (2.94 + 5.10i)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

−10.02468921750897213226898855822, −9.877957212312692523945181371867, −8.510789710198369306516934091284, −7.69066754188397852718192812771, −7.21020069800876697642296604161, −5.88610601254053390732483677958, −5.12428513109502511531692792234, −4.04777711705201069131632371100, −3.00151211257813740737618810884, −1.65030309417744172783924293998, 0.37981173124376751830964265212, 2.28432816658805819803791747962, 3.32342004773158713377004769434, 4.47198010119208306927467564107, 5.42853341579840920350056730167, 6.31888514369639979607512170558, 7.42129307474540474511439161870, 7.989907123332332146607987413331, 9.068381679592105364161473617464, 9.680335358394015971167410870514

Graph of the ZZ-function along the critical line