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Properties

Label 2-396-11.4-c3-0-9
Degree 22
Conductor 396396
Sign 0.905+0.424i0.905 + 0.424i
Analytic cond. 23.364723.3647
Root an. cond. 4.833714.83371
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.7 − 9.98i)5-s + (1.14 + 3.53i)7-s + (35.9 − 6.03i)11-s + (50.6 + 36.8i)13-s + (−67.4 + 49.0i)17-s + (15.3 − 47.3i)19-s + 42.7·23-s + (50.6 − 155. i)25-s + (−47.8 − 147. i)29-s + (163. + 118. i)31-s + (51.1 + 37.1i)35-s + (29.6 + 91.2i)37-s + (139. − 428. i)41-s − 343.·43-s + (−121. + 373. i)47-s + ⋯
L(s)  = 1  + (1.22 − 0.893i)5-s + (0.0620 + 0.190i)7-s + (0.986 − 0.165i)11-s + (1.08 + 0.785i)13-s + (−0.962 + 0.699i)17-s + (0.185 − 0.571i)19-s + 0.387·23-s + (0.404 − 1.24i)25-s + (−0.306 − 0.943i)29-s + (0.947 + 0.688i)31-s + (0.246 + 0.179i)35-s + (0.131 + 0.405i)37-s + (0.529 − 1.63i)41-s − 1.21·43-s + (−0.376 + 1.15i)47-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=((0.905+0.424i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+3/2) L(s)=((0.905+0.424i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: 0.905+0.424i0.905 + 0.424i
Analytic conductor: 23.364723.3647
Root analytic conductor: 4.833714.83371
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ396(37,⋅)\chi_{396} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 396, ( :3/2), 0.905+0.424i)(2,\ 396,\ (\ :3/2),\ 0.905 + 0.424i)

Particular Values

L(2)L(2) ≈\approx 2.6550675552.655067555
L(12)L(\frac12) ≈\approx 2.6550675552.655067555
L(52)L(\frac{5}{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
11 1+(−35.9+6.03i)T 1 + (-35.9 + 6.03i)T
good5 1+(−13.7+9.98i)T+(38.6−118.i)T2 1 + (-13.7 + 9.98i)T + (38.6 - 118. i)T^{2}
7 1+(−1.14−3.53i)T+(−277.+201.i)T2 1 + (-1.14 - 3.53i)T + (-277. + 201. i)T^{2}
13 1+(−50.6−36.8i)T+(678.+2.08e3i)T2 1 + (-50.6 - 36.8i)T + (678. + 2.08e3i)T^{2}
17 1+(67.4−49.0i)T+(1.51e3−4.67e3i)T2 1 + (67.4 - 49.0i)T + (1.51e3 - 4.67e3i)T^{2}
19 1+(−15.3+47.3i)T+(−5.54e3−4.03e3i)T2 1 + (-15.3 + 47.3i)T + (-5.54e3 - 4.03e3i)T^{2}
23 1−42.7T+1.21e4T2 1 - 42.7T + 1.21e4T^{2}
29 1+(47.8+147.i)T+(−1.97e4+1.43e4i)T2 1 + (47.8 + 147. i)T + (-1.97e4 + 1.43e4i)T^{2}
31 1+(−163.−118.i)T+(9.20e3+2.83e4i)T2 1 + (-163. - 118. i)T + (9.20e3 + 2.83e4i)T^{2}
37 1+(−29.6−91.2i)T+(−4.09e4+2.97e4i)T2 1 + (-29.6 - 91.2i)T + (-4.09e4 + 2.97e4i)T^{2}
41 1+(−139.+428.i)T+(−5.57e4−4.05e4i)T2 1 + (-139. + 428. i)T + (-5.57e4 - 4.05e4i)T^{2}
43 1+343.T+7.95e4T2 1 + 343.T + 7.95e4T^{2}
47 1+(121.−373.i)T+(−8.39e4−6.10e4i)T2 1 + (121. - 373. i)T + (-8.39e4 - 6.10e4i)T^{2}
53 1+(155.+113.i)T+(4.60e4+1.41e5i)T2 1 + (155. + 113. i)T + (4.60e4 + 1.41e5i)T^{2}
59 1+(−163.−503.i)T+(−1.66e5+1.20e5i)T2 1 + (-163. - 503. i)T + (-1.66e5 + 1.20e5i)T^{2}
61 1+(−273.+198.i)T+(7.01e4−2.15e5i)T2 1 + (-273. + 198. i)T + (7.01e4 - 2.15e5i)T^{2}
67 1−831.T+3.00e5T2 1 - 831.T + 3.00e5T^{2}
71 1+(−546.+397.i)T+(1.10e5−3.40e5i)T2 1 + (-546. + 397. i)T + (1.10e5 - 3.40e5i)T^{2}
73 1+(375.+1.15e3i)T+(−3.14e5+2.28e5i)T2 1 + (375. + 1.15e3i)T + (-3.14e5 + 2.28e5i)T^{2}
79 1+(−867.−630.i)T+(1.52e5+4.68e5i)T2 1 + (-867. - 630. i)T + (1.52e5 + 4.68e5i)T^{2}
83 1+(65.4−47.5i)T+(1.76e5−5.43e5i)T2 1 + (65.4 - 47.5i)T + (1.76e5 - 5.43e5i)T^{2}
89 1−30.9T+7.04e5T2 1 - 30.9T + 7.04e5T^{2}
97 1+(73.7+53.5i)T+(2.82e5+8.68e5i)T2 1 + (73.7 + 53.5i)T + (2.82e5 + 8.68e5i)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

−10.83360021716887366277270494519, −9.651574286157546454806131340180, −8.987477115217376092559460972881, −8.432943406430149982162140194770, −6.68178629006389395717312831916, −6.11329511841459377378808701068, −4.97943970963629531033501170766, −3.90848551426126947406928322720, −2.12333709562105516026266869217, −1.12105785736983865040003794802, 1.26572061257292023467256296032, 2.58858253386865127187929959712, 3.77658799111766520498093738315, 5.27305910877092778261403336806, 6.33604076593489567960417299618, 6.84771376418211397237195691894, 8.217314869105359288318075677097, 9.304347156860848764044655064374, 10.00073308609013382348173306448, 10.90532425270791541765516366371

Graph of the ZZ-function along the critical line