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Properties

Label 2-308-308.263-c1-0-36
Degree 22
Conductor 308308
Sign −0.323+0.946i-0.323 + 0.946i
Analytic cond. 2.459392.45939
Root an. cond. 1.568241.56824
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  + (1.39 − 0.228i)2-s + (−2.29 − 1.32i)3-s + (1.89 − 0.637i)4-s + (−3.5 − 1.32i)6-s + (0.5 − 2.59i)7-s + (2.49 − 1.32i)8-s + (2 + 3.46i)9-s + (−3.29 + 0.409i)11-s + (−5.18 − 1.04i)12-s − 2.64i·13-s + (0.104 − 3.74i)14-s + (3.18 − 2.41i)16-s + (−4.58 − 2.64i)17-s + (3.58 + 4.37i)18-s + (−4.58 + 5.29i)21-s + (−4.5 + 1.32i)22-s + ⋯
L(s)  = 1  + (0.986 − 0.161i)2-s + (−1.32 − 0.763i)3-s + (0.947 − 0.318i)4-s + (−1.42 − 0.540i)6-s + (0.188 − 0.981i)7-s + (0.883 − 0.467i)8-s + (0.666 + 1.15i)9-s + (−0.992 + 0.123i)11-s + (−1.49 − 0.302i)12-s − 0.733i·13-s + (0.0278 − 0.999i)14-s + (0.796 − 0.604i)16-s + (−1.11 − 0.641i)17-s + (0.844 + 1.03i)18-s + (−0.999 + 1.15i)21-s + (−0.959 + 0.282i)22-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((−0.323+0.946i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((−0.323+0.946i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: −0.323+0.946i-0.323 + 0.946i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(263,⋅)\chi_{308} (263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), −0.323+0.946i)(2,\ 308,\ (\ :1/2),\ -0.323 + 0.946i)

Particular Values

L(1)L(1) ≈\approx 0.855331−1.19575i0.855331 - 1.19575i
L(12)L(\frac12) ≈\approx 0.855331−1.19575i0.855331 - 1.19575i
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+(−1.39+0.228i)T 1 + (-1.39 + 0.228i)T
7 1+(−0.5+2.59i)T 1 + (-0.5 + 2.59i)T
11 1+(3.29−0.409i)T 1 + (3.29 - 0.409i)T
good3 1+(2.29+1.32i)T+(1.5+2.59i)T2 1 + (2.29 + 1.32i)T + (1.5 + 2.59i)T^{2}
5 1+(−2.5+4.33i)T2 1 + (-2.5 + 4.33i)T^{2}
13 1+2.64iT−13T2 1 + 2.64iT - 13T^{2}
17 1+(4.58+2.64i)T+(8.5+14.7i)T2 1 + (4.58 + 2.64i)T + (8.5 + 14.7i)T^{2}
19 1+(−9.5+16.4i)T2 1 + (-9.5 + 16.4i)T^{2}
23 1+(−4.58+2.64i)T+(11.5−19.9i)T2 1 + (-4.58 + 2.64i)T + (11.5 - 19.9i)T^{2}
29 1−7.93iT−29T2 1 - 7.93iT - 29T^{2}
31 1+(−9.16−5.29i)T+(15.5+26.8i)T2 1 + (-9.16 - 5.29i)T + (15.5 + 26.8i)T^{2}
37 1+(−4−6.92i)T+(−18.5+32.0i)T2 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2}
41 1−41T2 1 - 41T^{2}
43 1+2T+43T2 1 + 2T + 43T^{2}
47 1+(23.5−40.7i)T2 1 + (23.5 - 40.7i)T^{2}
53 1+(−2+3.46i)T+(−26.5−45.8i)T2 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2}
59 1+(−2.29−1.32i)T+(29.5+51.0i)T2 1 + (-2.29 - 1.32i)T + (29.5 + 51.0i)T^{2}
61 1+(−2.29+1.32i)T+(30.5−52.8i)T2 1 + (-2.29 + 1.32i)T + (30.5 - 52.8i)T^{2}
67 1+(−2.29−1.32i)T+(33.5+58.0i)T2 1 + (-2.29 - 1.32i)T + (33.5 + 58.0i)T^{2}
71 1−5.29iT−71T2 1 - 5.29iT - 71T^{2}
73 1+(−9.16−5.29i)T+(36.5+63.2i)T2 1 + (-9.16 - 5.29i)T + (36.5 + 63.2i)T^{2}
79 1+(6.5+11.2i)T+(−39.5+68.4i)T2 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(−7−12.1i)T+(−44.5+77.0i)T2 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2}
97 1−7T+97T2 1 - 7T + 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

−11.49624961309410699653524466650, −10.73767802608456938253703344607, −10.27698096812690365003583424580, −8.149163589166907966429394652271, −6.95883042136788658418464297997, −6.60143210499936082991329110987, −5.19063206548006622893456863828, −4.68596983327481833361181970022, −2.82184770882660956985849675235, −0.952084653986474754354346535006, 2.46110618836448759652781465770, 4.17924269116382637531296746780, 5.02621224051142506477212896199, 5.80909824950624064826701147276, 6.58875258456860559200844622730, 8.003166608613677448862360387990, 9.351299076186921942606459168651, 10.56368031646566724887172228355, 11.31944622142821509622600068676, 11.74909369496926883556991457824

Graph of the ZZ-function along the critical line