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Properties

Label 2-396-11.3-c3-0-2
Degree 22
Conductor 396396
Sign −0.826−0.562i-0.826 - 0.562i
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  + (10.2 + 7.42i)5-s + (−9.75 + 30.0i)7-s + (13.1 + 34.0i)11-s + (−3.98 + 2.89i)13-s + (−36.6 − 26.5i)17-s + (−10.6 − 32.6i)19-s − 206.·23-s + (10.6 + 32.7i)25-s + (11.0 − 34.1i)29-s + (18.0 − 13.0i)31-s + (−322. + 234. i)35-s + (58.9 − 181. i)37-s + (18.5 + 56.9i)41-s + 334.·43-s + (101. + 313. i)47-s + ⋯
L(s)  = 1  + (0.913 + 0.663i)5-s + (−0.526 + 1.62i)7-s + (0.360 + 0.932i)11-s + (−0.0850 + 0.0617i)13-s + (−0.522 − 0.379i)17-s + (−0.128 − 0.394i)19-s − 1.87·23-s + (0.0851 + 0.262i)25-s + (0.0710 − 0.218i)29-s + (0.104 − 0.0758i)31-s + (−1.55 + 1.13i)35-s + (0.262 − 0.806i)37-s + (0.0705 + 0.217i)41-s + 1.18·43-s + (0.315 + 0.971i)47-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=((−0.826−0.562i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+3/2) L(s)=((−0.826−0.562i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: −0.826−0.562i-0.826 - 0.562i
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), −0.826−0.562i)(2,\ 396,\ (\ :3/2),\ -0.826 - 0.562i)

Particular Values

L(2)L(2) ≈\approx 1.4395270451.439527045
L(12)L(\frac12) ≈\approx 1.4395270451.439527045
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+(−13.1−34.0i)T 1 + (-13.1 - 34.0i)T
good5 1+(−10.2−7.42i)T+(38.6+118.i)T2 1 + (-10.2 - 7.42i)T + (38.6 + 118. i)T^{2}
7 1+(9.75−30.0i)T+(−277.−201.i)T2 1 + (9.75 - 30.0i)T + (-277. - 201. i)T^{2}
13 1+(3.98−2.89i)T+(678.−2.08e3i)T2 1 + (3.98 - 2.89i)T + (678. - 2.08e3i)T^{2}
17 1+(36.6+26.5i)T+(1.51e3+4.67e3i)T2 1 + (36.6 + 26.5i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(10.6+32.6i)T+(−5.54e3+4.03e3i)T2 1 + (10.6 + 32.6i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1+206.T+1.21e4T2 1 + 206.T + 1.21e4T^{2}
29 1+(−11.0+34.1i)T+(−1.97e4−1.43e4i)T2 1 + (-11.0 + 34.1i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(−18.0+13.0i)T+(9.20e3−2.83e4i)T2 1 + (-18.0 + 13.0i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(−58.9+181.i)T+(−4.09e4−2.97e4i)T2 1 + (-58.9 + 181. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(−18.5−56.9i)T+(−5.57e4+4.05e4i)T2 1 + (-18.5 - 56.9i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1−334.T+7.95e4T2 1 - 334.T + 7.95e4T^{2}
47 1+(−101.−313.i)T+(−8.39e4+6.10e4i)T2 1 + (-101. - 313. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(296.−215.i)T+(4.60e4−1.41e5i)T2 1 + (296. - 215. i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(207.−637.i)T+(−1.66e5−1.20e5i)T2 1 + (207. - 637. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−35.5−25.8i)T+(7.01e4+2.15e5i)T2 1 + (-35.5 - 25.8i)T + (7.01e4 + 2.15e5i)T^{2}
67 1−673.T+3.00e5T2 1 - 673.T + 3.00e5T^{2}
71 1+(738.+536.i)T+(1.10e5+3.40e5i)T2 1 + (738. + 536. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(364.−1.12e3i)T+(−3.14e5−2.28e5i)T2 1 + (364. - 1.12e3i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(566.−411.i)T+(1.52e5−4.68e5i)T2 1 + (566. - 411. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(−391.−284.i)T+(1.76e5+5.43e5i)T2 1 + (-391. - 284. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1+1.22e3T+7.04e5T2 1 + 1.22e3T + 7.04e5T^{2}
97 1+(−806.+586.i)T+(2.82e5−8.68e5i)T2 1 + (-806. + 586. 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

−11.29364302435595974129230947527, −10.06535246661738966420755231756, −9.519337802615920262399636813129, −8.733910342516855825366200126613, −7.37537727294118869749588902251, −6.23707265187408196409242846922, −5.80832049156531593859249742643, −4.37864285256927220865548450161, −2.67486640682902430245019262617, −2.07982617699353144265824181926, 0.45588299706617171934603870587, 1.71196006007532334526287085813, 3.51064696333864638028004486490, 4.43775761865380178030613683034, 5.82886514713633352439434809346, 6.53523863468216408748244667264, 7.73644820392547710895181793892, 8.713420830487203927845618505251, 9.767500945531203463503843102990, 10.31272495381004476962095552226

Graph of the ZZ-function along the critical line