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Properties

Label 2-396-11.3-c3-0-8
Degree 22
Conductor 396396
Sign 0.739+0.673i0.739 + 0.673i
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  + (−1.60 − 1.16i)5-s + (3.78 − 11.6i)7-s + (8.21 + 35.5i)11-s + (5.26 − 3.82i)13-s + (62.9 + 45.7i)17-s + (−28.2 − 86.9i)19-s + 20.2·23-s + (−37.4 − 115. i)25-s + (23.5 − 72.5i)29-s + (92.8 − 67.4i)31-s + (−19.6 + 14.2i)35-s + (27.6 − 85.0i)37-s + (−30.4 − 93.7i)41-s + 488.·43-s + (−33.0 − 101. i)47-s + ⋯
L(s)  = 1  + (−0.143 − 0.104i)5-s + (0.204 − 0.629i)7-s + (0.225 + 0.974i)11-s + (0.112 − 0.0816i)13-s + (0.898 + 0.652i)17-s + (−0.341 − 1.05i)19-s + 0.183·23-s + (−0.299 − 0.921i)25-s + (0.151 − 0.464i)29-s + (0.538 − 0.390i)31-s + (−0.0950 + 0.0690i)35-s + (0.122 − 0.377i)37-s + (−0.116 − 0.357i)41-s + 1.73·43-s + (−0.102 − 0.315i)47-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=((0.739+0.673i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+3/2) L(s)=((0.739+0.673i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: 0.739+0.673i0.739 + 0.673i
Analytic conductor: 23.364723.3647
Root analytic conductor: 4.833714.83371
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ396(289,⋅)\chi_{396} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 396, ( :3/2), 0.739+0.673i)(2,\ 396,\ (\ :3/2),\ 0.739 + 0.673i)

Particular Values

L(2)L(2) ≈\approx 1.8593432151.859343215
L(12)L(\frac12) ≈\approx 1.8593432151.859343215
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+(−8.21−35.5i)T 1 + (-8.21 - 35.5i)T
good5 1+(1.60+1.16i)T+(38.6+118.i)T2 1 + (1.60 + 1.16i)T + (38.6 + 118. i)T^{2}
7 1+(−3.78+11.6i)T+(−277.−201.i)T2 1 + (-3.78 + 11.6i)T + (-277. - 201. i)T^{2}
13 1+(−5.26+3.82i)T+(678.−2.08e3i)T2 1 + (-5.26 + 3.82i)T + (678. - 2.08e3i)T^{2}
17 1+(−62.9−45.7i)T+(1.51e3+4.67e3i)T2 1 + (-62.9 - 45.7i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(28.2+86.9i)T+(−5.54e3+4.03e3i)T2 1 + (28.2 + 86.9i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1−20.2T+1.21e4T2 1 - 20.2T + 1.21e4T^{2}
29 1+(−23.5+72.5i)T+(−1.97e4−1.43e4i)T2 1 + (-23.5 + 72.5i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(−92.8+67.4i)T+(9.20e3−2.83e4i)T2 1 + (-92.8 + 67.4i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(−27.6+85.0i)T+(−4.09e4−2.97e4i)T2 1 + (-27.6 + 85.0i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(30.4+93.7i)T+(−5.57e4+4.05e4i)T2 1 + (30.4 + 93.7i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1−488.T+7.95e4T2 1 - 488.T + 7.95e4T^{2}
47 1+(33.0+101.i)T+(−8.39e4+6.10e4i)T2 1 + (33.0 + 101. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(−447.+324.i)T+(4.60e4−1.41e5i)T2 1 + (-447. + 324. i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(−52.3+161.i)T+(−1.66e5−1.20e5i)T2 1 + (-52.3 + 161. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−235.−170.i)T+(7.01e4+2.15e5i)T2 1 + (-235. - 170. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1−107.T+3.00e5T2 1 - 107.T + 3.00e5T^{2}
71 1+(589.+428.i)T+(1.10e5+3.40e5i)T2 1 + (589. + 428. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−110.+341.i)T+(−3.14e5−2.28e5i)T2 1 + (-110. + 341. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(−413.+300.i)T+(1.52e5−4.68e5i)T2 1 + (-413. + 300. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(515.+374.i)T+(1.76e5+5.43e5i)T2 1 + (515. + 374. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1−830.T+7.04e5T2 1 - 830.T + 7.04e5T^{2}
97 1+(642.−467.i)T+(2.82e5−8.68e5i)T2 1 + (642. - 467. i)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

−10.64698210090855686028115557192, −9.977941697919814988837592126884, −8.931657535555462430069002173338, −7.88021338924552007372743118742, −7.10955616820058866935952901312, −6.01324774474976655173638878206, −4.70078501917344611261415534163, −3.88481125281832366569013716494, −2.29489334062964043914298022113, −0.76131441691442274223936093154, 1.15114567232830177157069638460, 2.77495908305154234406702210301, 3.87256049378878329032139221981, 5.32119394813441900224262993848, 6.05438210538410741102650651704, 7.30413103372518953802099801891, 8.279756740225235643700434412514, 9.055329774408811844430255306596, 10.07751344902050476034833667059, 11.07537996400023164646926714509

Graph of the ZZ-function along the critical line