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Properties

Label 2-936-13.9-c1-0-12
Degree 22
Conductor 936936
Sign −0.575+0.818i-0.575 + 0.818i
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  − 3.17·5-s + (1.08 + 1.88i)7-s + (1.45 − 2.52i)11-s + (−1.21 + 3.39i)13-s + (−3.04 − 5.27i)17-s + (−1.45 − 2.52i)19-s + (0.281 − 0.488i)23-s + 5.09·25-s + (1.32 − 2.30i)29-s − 7.09·31-s + (−3.45 − 5.99i)35-s + (3.76 − 6.52i)37-s + (−1.30 + 2.26i)41-s + (−5.00 − 8.67i)43-s − 3.43·47-s + ⋯
L(s)  = 1  − 1.42·5-s + (0.411 + 0.712i)7-s + (0.439 − 0.762i)11-s + (−0.337 + 0.941i)13-s + (−0.739 − 1.28i)17-s + (−0.334 − 0.579i)19-s + (0.0587 − 0.101i)23-s + 1.01·25-s + (0.246 − 0.427i)29-s − 1.27·31-s + (−0.584 − 1.01i)35-s + (0.619 − 1.07i)37-s + (−0.204 + 0.353i)41-s + (−0.763 − 1.32i)43-s − 0.501·47-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((−0.575+0.818i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((−0.575+0.818i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: −0.575+0.818i-0.575 + 0.818i
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), −0.575+0.818i)(2,\ 936,\ (\ :1/2),\ -0.575 + 0.818i)

Particular Values

L(1)L(1) ≈\approx 0.228471−0.439953i0.228471 - 0.439953i
L(12)L(\frac12) ≈\approx 0.228471−0.439953i0.228471 - 0.439953i
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+(1.21−3.39i)T 1 + (1.21 - 3.39i)T
good5 1+3.17T+5T2 1 + 3.17T + 5T^{2}
7 1+(−1.08−1.88i)T+(−3.5+6.06i)T2 1 + (-1.08 - 1.88i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1.45+2.52i)T+(−5.5−9.52i)T2 1 + (-1.45 + 2.52i)T + (-5.5 - 9.52i)T^{2}
17 1+(3.04+5.27i)T+(−8.5+14.7i)T2 1 + (3.04 + 5.27i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.45+2.52i)T+(−9.5+16.4i)T2 1 + (1.45 + 2.52i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.281+0.488i)T+(−11.5−19.9i)T2 1 + (-0.281 + 0.488i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.32+2.30i)T+(−14.5−25.1i)T2 1 + (-1.32 + 2.30i)T + (-14.5 - 25.1i)T^{2}
31 1+7.09T+31T2 1 + 7.09T + 31T^{2}
37 1+(−3.76+6.52i)T+(−18.5−32.0i)T2 1 + (-3.76 + 6.52i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.30−2.26i)T+(−20.5−35.5i)T2 1 + (1.30 - 2.26i)T + (-20.5 - 35.5i)T^{2}
43 1+(5.00+8.67i)T+(−21.5+37.2i)T2 1 + (5.00 + 8.67i)T + (-21.5 + 37.2i)T^{2}
47 1+3.43T+47T2 1 + 3.43T + 47T^{2}
53 1+7.17T+53T2 1 + 7.17T + 53T^{2}
59 1+(7.27+12.5i)T+(−29.5+51.0i)T2 1 + (7.27 + 12.5i)T + (-29.5 + 51.0i)T^{2}
61 1+(4.39+7.61i)T+(−30.5+52.8i)T2 1 + (4.39 + 7.61i)T + (-30.5 + 52.8i)T^{2}
67 1+(5.52−9.56i)T+(−33.5−58.0i)T2 1 + (5.52 - 9.56i)T + (-33.5 - 58.0i)T^{2}
71 1+(−3.71−6.44i)T+(−35.5+61.4i)T2 1 + (-3.71 - 6.44i)T + (-35.5 + 61.4i)T^{2}
73 1−13.1T+73T2 1 - 13.1T + 73T^{2}
79 1−9.96T+79T2 1 - 9.96T + 79T^{2}
83 1+13.7T+83T2 1 + 13.7T + 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+(0.629+1.09i)T+(−48.5+84.0i)T2 1 + (0.629 + 1.09i)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

−9.488875644045103387490732202307, −8.891803169913963078626245647077, −8.161894569852843207640565549656, −7.24571930903093917921320494806, −6.52477706257969478525316259903, −5.20085646178555306171078709310, −4.39833149707529973828913761754, −3.44803597520639179846382263660, −2.19533440453654932568018702070, −0.23621612107437885500248268702, 1.52509614412317276971244467712, 3.25816119673642692956014146968, 4.15641709084642720352525109441, 4.75860797723116956804463794867, 6.18179989769101395374354511135, 7.18358582965972441486639306058, 7.85674745713607124943973021352, 8.390424871007786359746725847815, 9.546323828273021323855284000855, 10.62013713752113578352264560418

Graph of the ZZ-function along the critical line