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Properties

Label 2-936-39.32-c1-0-4
Degree 22
Conductor 936936
Sign 0.177−0.984i0.177 - 0.984i
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  + (2.43 + 2.43i)5-s + (−0.0623 + 0.232i)7-s + (0.655 + 2.44i)11-s + (0.765 + 3.52i)13-s + (0.156 + 0.270i)17-s + (−4.72 − 1.26i)19-s + (1.56 − 2.70i)23-s + 6.84i·25-s + (0.251 + 0.145i)29-s + (−2.38 + 2.38i)31-s + (−0.717 + 0.414i)35-s + (−2.72 + 0.730i)37-s + (5.24 − 1.40i)41-s + (−7.61 + 4.39i)43-s + (9.47 − 9.47i)47-s + ⋯
L(s)  = 1  + (1.08 + 1.08i)5-s + (−0.0235 + 0.0878i)7-s + (0.197 + 0.737i)11-s + (0.212 + 0.977i)13-s + (0.0378 + 0.0655i)17-s + (−1.08 − 0.290i)19-s + (0.326 − 0.565i)23-s + 1.36i·25-s + (0.0466 + 0.0269i)29-s + (−0.427 + 0.427i)31-s + (−0.121 + 0.0700i)35-s + (−0.448 + 0.120i)37-s + (0.819 − 0.219i)41-s + (−1.16 + 0.670i)43-s + (1.38 − 1.38i)47-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((0.177−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((0.177−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 0.177−0.984i0.177 - 0.984i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(305,⋅)\chi_{936} (305, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), 0.177−0.984i)(2,\ 936,\ (\ :1/2),\ 0.177 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 1.38918+1.16119i1.38918 + 1.16119i
L(12)L(\frac12) ≈\approx 1.38918+1.16119i1.38918 + 1.16119i
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+(−0.765−3.52i)T 1 + (-0.765 - 3.52i)T
good5 1+(−2.43−2.43i)T+5iT2 1 + (-2.43 - 2.43i)T + 5iT^{2}
7 1+(0.0623−0.232i)T+(−6.06−3.5i)T2 1 + (0.0623 - 0.232i)T + (-6.06 - 3.5i)T^{2}
11 1+(−0.655−2.44i)T+(−9.52+5.5i)T2 1 + (-0.655 - 2.44i)T + (-9.52 + 5.5i)T^{2}
17 1+(−0.156−0.270i)T+(−8.5+14.7i)T2 1 + (-0.156 - 0.270i)T + (-8.5 + 14.7i)T^{2}
19 1+(4.72+1.26i)T+(16.4+9.5i)T2 1 + (4.72 + 1.26i)T + (16.4 + 9.5i)T^{2}
23 1+(−1.56+2.70i)T+(−11.5−19.9i)T2 1 + (-1.56 + 2.70i)T + (-11.5 - 19.9i)T^{2}
29 1+(−0.251−0.145i)T+(14.5+25.1i)T2 1 + (-0.251 - 0.145i)T + (14.5 + 25.1i)T^{2}
31 1+(2.38−2.38i)T−31iT2 1 + (2.38 - 2.38i)T - 31iT^{2}
37 1+(2.72−0.730i)T+(32.0−18.5i)T2 1 + (2.72 - 0.730i)T + (32.0 - 18.5i)T^{2}
41 1+(−5.24+1.40i)T+(35.5−20.5i)T2 1 + (-5.24 + 1.40i)T + (35.5 - 20.5i)T^{2}
43 1+(7.61−4.39i)T+(21.5−37.2i)T2 1 + (7.61 - 4.39i)T + (21.5 - 37.2i)T^{2}
47 1+(−9.47+9.47i)T−47iT2 1 + (-9.47 + 9.47i)T - 47iT^{2}
53 1−7.02iT−53T2 1 - 7.02iT - 53T^{2}
59 1+(4.77+1.28i)T+(51.0+29.5i)T2 1 + (4.77 + 1.28i)T + (51.0 + 29.5i)T^{2}
61 1+(−1.10−1.91i)T+(−30.5+52.8i)T2 1 + (-1.10 - 1.91i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.0954+0.356i)T+(−58.0+33.5i)T2 1 + (0.0954 + 0.356i)T + (-58.0 + 33.5i)T^{2}
71 1+(−2.10+7.84i)T+(−61.4−35.5i)T2 1 + (-2.10 + 7.84i)T + (-61.4 - 35.5i)T^{2}
73 1+(11.1+11.1i)T+73iT2 1 + (11.1 + 11.1i)T + 73iT^{2}
79 1−9.82T+79T2 1 - 9.82T + 79T^{2}
83 1+(−10.6−10.6i)T+83iT2 1 + (-10.6 - 10.6i)T + 83iT^{2}
89 1+(−3.43−12.8i)T+(−77.0+44.5i)T2 1 + (-3.43 - 12.8i)T + (-77.0 + 44.5i)T^{2}
97 1+(−10.7−2.86i)T+(84.0+48.5i)T2 1 + (-10.7 - 2.86i)T + (84.0 + 48.5i)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

−10.47634659534129371288254751303, −9.364496036092380009292077309196, −8.871786149116413435761674405187, −7.49808050755208648464395667501, −6.64531515087820503252076409179, −6.24188819819334776774013883702, −5.02508196931163738408796156946, −3.93387266707176818505828712689, −2.60722818423463892706339148879, −1.82175507435275589396155163765, 0.870297577756179951669129821198, 2.11433310989282674575220159123, 3.49741776876675517727243917886, 4.69688618392329317999597022590, 5.66690608818552736529395091612, 6.08640910472133363742004158945, 7.40662173334387907515932894452, 8.500104753053459921737894616993, 8.914528186968929046243254537523, 9.861835157637459501183411230748

Graph of the ZZ-function along the critical line