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Properties

Label 2-308-308.195-c2-0-33
Degree 22
Conductor 308308
Sign −0.218−0.975i-0.218 - 0.975i
Analytic cond. 8.392398.39239
Root an. cond. 2.896962.89696
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.72 + 1.01i)2-s + (1.92 + 3.50i)4-s + (−2.16 + 6.65i)7-s + (−0.247 + 7.99i)8-s + (7.28 − 5.29i)9-s + (−3.79 + 10.3i)11-s + (−10.5 + 9.26i)14-s + (−8.56 + 13.5i)16-s + (17.9 − 1.69i)18-s + (−17.0 + 13.9i)22-s + 4.03i·23-s + (7.72 + 23.7i)25-s + (−27.5 + 5.25i)28-s + (13.9 + 4.51i)29-s + (−28.5 + 14.5i)32-s + ⋯
L(s)  = 1  + (0.860 + 0.508i)2-s + (0.482 + 0.876i)4-s + (−0.309 + 0.951i)7-s + (−0.0309 + 0.999i)8-s + (0.809 − 0.587i)9-s + (−0.344 + 0.938i)11-s + (−0.750 + 0.661i)14-s + (−0.535 + 0.844i)16-s + (0.995 − 0.0942i)18-s + (−0.774 + 0.632i)22-s + 0.175i·23-s + (0.309 + 0.951i)25-s + (−0.982 + 0.187i)28-s + (0.479 + 0.155i)29-s + (−0.890 + 0.454i)32-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((−0.218−0.975i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1) L(s)=((−0.218−0.975i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: −0.218−0.975i-0.218 - 0.975i
Analytic conductor: 8.392398.39239
Root analytic conductor: 2.896962.89696
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ308(195,⋅)\chi_{308} (195, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1), −0.218−0.975i)(2,\ 308,\ (\ :1),\ -0.218 - 0.975i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.61081+2.01170i1.61081 + 2.01170i
L(12)L(\frac12) ≈\approx 1.61081+2.01170i1.61081 + 2.01170i
L(2)L(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+(−1.72−1.01i)T 1 + (-1.72 - 1.01i)T
7 1+(2.16−6.65i)T 1 + (2.16 - 6.65i)T
11 1+(3.79−10.3i)T 1 + (3.79 - 10.3i)T
good3 1+(−7.28+5.29i)T2 1 + (-7.28 + 5.29i)T^{2}
5 1+(−7.72−23.7i)T2 1 + (-7.72 - 23.7i)T^{2}
13 1+(52.2−160.i)T2 1 + (52.2 - 160. i)T^{2}
17 1+(89.3+274.i)T2 1 + (89.3 + 274. i)T^{2}
19 1+(292.−212.i)T2 1 + (292. - 212. i)T^{2}
23 1−4.03iT−529T2 1 - 4.03iT - 529T^{2}
29 1+(−13.9−4.51i)T+(680.+494.i)T2 1 + (-13.9 - 4.51i)T + (680. + 494. i)T^{2}
31 1+(296.−913.i)T2 1 + (296. - 913. i)T^{2}
37 1+(−22.2+68.6i)T+(−1.10e3−804.i)T2 1 + (-22.2 + 68.6i)T + (-1.10e3 - 804. i)T^{2}
41 1+(−1.35e3+988.i)T2 1 + (-1.35e3 + 988. i)T^{2}
43 1−9.59T+1.84e3T2 1 - 9.59T + 1.84e3T^{2}
47 1+(−1.78e3+1.29e3i)T2 1 + (-1.78e3 + 1.29e3i)T^{2}
53 1+(−54.2+39.4i)T+(868.−2.67e3i)T2 1 + (-54.2 + 39.4i)T + (868. - 2.67e3i)T^{2}
59 1+(−2.81e3−2.04e3i)T2 1 + (-2.81e3 - 2.04e3i)T^{2}
61 1+(1.14e3+3.53e3i)T2 1 + (1.14e3 + 3.53e3i)T^{2}
67 1+17.9iT−4.48e3T2 1 + 17.9iT - 4.48e3T^{2}
71 1+(−79.6+109.i)T+(−1.55e3−4.79e3i)T2 1 + (-79.6 + 109. i)T + (-1.55e3 - 4.79e3i)T^{2}
73 1+(−4.31e3−3.13e3i)T2 1 + (-4.31e3 - 3.13e3i)T^{2}
79 1+(−1.13+0.823i)T+(1.92e3−5.93e3i)T2 1 + (-1.13 + 0.823i)T + (1.92e3 - 5.93e3i)T^{2}
83 1+(−2.12e3−6.55e3i)T2 1 + (-2.12e3 - 6.55e3i)T^{2}
89 1−7.92e3T2 1 - 7.92e3T^{2}
97 1+(−2.90e3+8.94e3i)T2 1 + (-2.90e3 + 8.94e3i)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

−12.18420055197837156057082821914, −11.04518002947137632267974206172, −9.753091485429978623287832863767, −8.871585311217378268224584676442, −7.59752578190093283129160058345, −6.80858462379032023695867926737, −5.75576683783093954822408554424, −4.74918458439529716216669103494, −3.55265313850399882143580943897, −2.18255204209033464938752976686, 1.00229200656210399053777468339, 2.72509536766783651161816015215, 3.98473790559794955034361093654, 4.88357602304156357927704856669, 6.18759787419397384278873016440, 7.09586502970071414671152088925, 8.252921057642354797701397236849, 9.821185728476640948568509013155, 10.43170945348004314564972838119, 11.15907508870110536347323217068

Graph of the ZZ-function along the critical line