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Properties

Label 2-3276-13.10-c1-0-25
Degree 22
Conductor 32763276
Sign 0.283+0.958i0.283 + 0.958i
Analytic cond. 26.158926.1589
Root an. cond. 5.114585.11458
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  + 1.91i·5-s + (0.866 + 0.5i)7-s + (−2.71 + 1.56i)11-s + (2.44 − 2.64i)13-s + (−1.02 + 1.78i)17-s + (−4.35 − 2.51i)19-s + (−3.46 − 6.00i)23-s + 1.32·25-s + (−5.03 − 8.71i)29-s − 10.5i·31-s + (−0.959 + 1.66i)35-s + (−0.508 + 0.293i)37-s + (−1.22 + 0.709i)41-s + (−1.39 + 2.41i)43-s + 3.70i·47-s + ⋯
L(s)  = 1  + 0.857i·5-s + (0.327 + 0.188i)7-s + (−0.819 + 0.472i)11-s + (0.679 − 0.734i)13-s + (−0.249 + 0.431i)17-s + (−0.998 − 0.576i)19-s + (−0.723 − 1.25i)23-s + 0.264·25-s + (−0.934 − 1.61i)29-s − 1.89i·31-s + (−0.162 + 0.280i)35-s + (−0.0835 + 0.0482i)37-s + (−0.191 + 0.110i)41-s + (−0.212 + 0.367i)43-s + 0.540i·47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((0.283+0.958i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((0.283+0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.283+0.958i0.283 + 0.958i
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3276(1765,⋅)\chi_{3276} (1765, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), 0.283+0.958i)(2,\ 3276,\ (\ :1/2),\ 0.283 + 0.958i)

Particular Values

L(1)L(1) ≈\approx 1.1487453951.148745395
L(12)L(\frac12) ≈\approx 1.1487453951.148745395
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
7 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1+(−2.44+2.64i)T 1 + (-2.44 + 2.64i)T
good5 1−1.91iT−5T2 1 - 1.91iT - 5T^{2}
11 1+(2.71−1.56i)T+(5.5−9.52i)T2 1 + (2.71 - 1.56i)T + (5.5 - 9.52i)T^{2}
17 1+(1.02−1.78i)T+(−8.5−14.7i)T2 1 + (1.02 - 1.78i)T + (-8.5 - 14.7i)T^{2}
19 1+(4.35+2.51i)T+(9.5+16.4i)T2 1 + (4.35 + 2.51i)T + (9.5 + 16.4i)T^{2}
23 1+(3.46+6.00i)T+(−11.5+19.9i)T2 1 + (3.46 + 6.00i)T + (-11.5 + 19.9i)T^{2}
29 1+(5.03+8.71i)T+(−14.5+25.1i)T2 1 + (5.03 + 8.71i)T + (-14.5 + 25.1i)T^{2}
31 1+10.5iT−31T2 1 + 10.5iT - 31T^{2}
37 1+(0.508−0.293i)T+(18.5−32.0i)T2 1 + (0.508 - 0.293i)T + (18.5 - 32.0i)T^{2}
41 1+(1.22−0.709i)T+(20.5−35.5i)T2 1 + (1.22 - 0.709i)T + (20.5 - 35.5i)T^{2}
43 1+(1.39−2.41i)T+(−21.5−37.2i)T2 1 + (1.39 - 2.41i)T + (-21.5 - 37.2i)T^{2}
47 1−3.70iT−47T2 1 - 3.70iT - 47T^{2}
53 1−12.4T+53T2 1 - 12.4T + 53T^{2}
59 1+(−3.93−2.27i)T+(29.5+51.0i)T2 1 + (-3.93 - 2.27i)T + (29.5 + 51.0i)T^{2}
61 1+(−1.20+2.08i)T+(−30.5−52.8i)T2 1 + (-1.20 + 2.08i)T + (-30.5 - 52.8i)T^{2}
67 1+(−12.7+7.33i)T+(33.5−58.0i)T2 1 + (-12.7 + 7.33i)T + (33.5 - 58.0i)T^{2}
71 1+(6.32+3.65i)T+(35.5+61.4i)T2 1 + (6.32 + 3.65i)T + (35.5 + 61.4i)T^{2}
73 1−2.18iT−73T2 1 - 2.18iT - 73T^{2}
79 1−0.00212T+79T2 1 - 0.00212T + 79T^{2}
83 1−9.23iT−83T2 1 - 9.23iT - 83T^{2}
89 1+(−12.3+7.12i)T+(44.5−77.0i)T2 1 + (-12.3 + 7.12i)T + (44.5 - 77.0i)T^{2}
97 1+(1.08+0.625i)T+(48.5+84.0i)T2 1 + (1.08 + 0.625i)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

−8.183274697643044424567408404588, −7.979067014090404948898955743454, −6.94994163148071810106375312140, −6.22216591172019134133341625190, −5.62153396847248070368631833735, −4.50866926966198205622740229679, −3.85181506598952364713662971237, −2.58556396090839606693180466999, −2.18122558740481675773803010492, −0.36077562533224782490899938757, 1.18461830890709477909926426467, 2.05052667464847936685450321158, 3.41026565831203392200005306598, 4.10284016618305827074012969772, 5.18511717955320488445138010130, 5.46386749899628048719733068646, 6.64039313024853596226808726781, 7.29168061336178818757894075684, 8.311843036763357139552918770456, 8.668024742083726655446218611703

Graph of the ZZ-function along the critical line