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Properties

Label 2-304-16.13-c1-0-13
Degree 22
Conductor 304304
Sign 0.807+0.589i0.807 + 0.589i
Analytic cond. 2.427452.42745
Root an. cond. 1.558021.55802
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  + (−1.33 + 0.462i)2-s + (0.227 + 0.227i)3-s + (1.57 − 1.23i)4-s + (−2.67 + 2.67i)5-s + (−0.409 − 0.198i)6-s − 4.39i·7-s + (−1.52 + 2.37i)8-s − 2.89i·9-s + (2.33 − 4.81i)10-s + (2.02 − 2.02i)11-s + (0.639 + 0.0762i)12-s + (1.66 + 1.66i)13-s + (2.03 + 5.87i)14-s − 1.21·15-s + (0.941 − 3.88i)16-s + 4.30·17-s + ⋯
L(s)  = 1  + (−0.944 + 0.327i)2-s + (0.131 + 0.131i)3-s + (0.785 − 0.618i)4-s + (−1.19 + 1.19i)5-s + (−0.167 − 0.0811i)6-s − 1.66i·7-s + (−0.540 + 0.841i)8-s − 0.965i·9-s + (0.739 − 1.52i)10-s + (0.610 − 0.610i)11-s + (0.184 + 0.0220i)12-s + (0.461 + 0.461i)13-s + (0.543 + 1.57i)14-s − 0.314·15-s + (0.235 − 0.971i)16-s + 1.04·17-s + ⋯

Functional equation

Λ(s)=(304s/2 ΓC(s) L(s)=((0.807+0.589i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(304s/2 ΓC(s+1/2) L(s)=((0.807+0.589i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 304304    =    24⋅192^{4} \cdot 19
Sign: 0.807+0.589i0.807 + 0.589i
Analytic conductor: 2.427452.42745
Root analytic conductor: 1.558021.55802
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ304(77,⋅)\chi_{304} (77, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 304, ( :1/2), 0.807+0.589i)(2,\ 304,\ (\ :1/2),\ 0.807 + 0.589i)

Particular Values

L(1)L(1) ≈\approx 0.665098−0.216823i0.665098 - 0.216823i
L(12)L(\frac12) ≈\approx 0.665098−0.216823i0.665098 - 0.216823i
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+(1.33−0.462i)T 1 + (1.33 - 0.462i)T
19 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good3 1+(−0.227−0.227i)T+3iT2 1 + (-0.227 - 0.227i)T + 3iT^{2}
5 1+(2.67−2.67i)T−5iT2 1 + (2.67 - 2.67i)T - 5iT^{2}
7 1+4.39iT−7T2 1 + 4.39iT - 7T^{2}
11 1+(−2.02+2.02i)T−11iT2 1 + (-2.02 + 2.02i)T - 11iT^{2}
13 1+(−1.66−1.66i)T+13iT2 1 + (-1.66 - 1.66i)T + 13iT^{2}
17 1−4.30T+17T2 1 - 4.30T + 17T^{2}
23 1+1.00iT−23T2 1 + 1.00iT - 23T^{2}
29 1+(3.73+3.73i)T+29iT2 1 + (3.73 + 3.73i)T + 29iT^{2}
31 1−10.1T+31T2 1 - 10.1T + 31T^{2}
37 1+(4.90−4.90i)T−37iT2 1 + (4.90 - 4.90i)T - 37iT^{2}
41 1+9.29iT−41T2 1 + 9.29iT - 41T^{2}
43 1+(−5.74+5.74i)T−43iT2 1 + (-5.74 + 5.74i)T - 43iT^{2}
47 1+4.84T+47T2 1 + 4.84T + 47T^{2}
53 1+(−3.75+3.75i)T−53iT2 1 + (-3.75 + 3.75i)T - 53iT^{2}
59 1+(3.38−3.38i)T−59iT2 1 + (3.38 - 3.38i)T - 59iT^{2}
61 1+(7.04+7.04i)T+61iT2 1 + (7.04 + 7.04i)T + 61iT^{2}
67 1+(1.62+1.62i)T+67iT2 1 + (1.62 + 1.62i)T + 67iT^{2}
71 1−2.97iT−71T2 1 - 2.97iT - 71T^{2}
73 1−13.7iT−73T2 1 - 13.7iT - 73T^{2}
79 1+5.25T+79T2 1 + 5.25T + 79T^{2}
83 1+(−2.32−2.32i)T+83iT2 1 + (-2.32 - 2.32i)T + 83iT^{2}
89 1+3.66iT−89T2 1 + 3.66iT - 89T^{2}
97 1−4.82T+97T2 1 - 4.82T + 97T^{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

−11.40416318938332047420741414170, −10.56767170263278010651378926742, −9.929101749325527634956919162374, −8.636961416938732352152601746131, −7.69259016051801518415617348053, −6.94114745524518471787417575963, −6.30527898482893774677561890351, −4.03983052805044543208024341959, −3.29179899463893022226200500454, −0.75801192015733620283767210215, 1.52892259209321896057918812715, 3.10001205439589619975837483572, 4.64961491095366895901744542176, 5.89187049976341771393078381839, 7.53746292113813337118743654648, 8.215462401513083592304835083306, 8.809917464355305724597867514236, 9.683467038773592313309365834020, 11.01260148840497143469392344244, 11.99380789042129745677229772556

Graph of the ZZ-function along the critical line