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Properties

Label 2-308-11.9-c1-0-3
Degree 22
Conductor 308308
Sign 0.957−0.288i0.957 - 0.288i
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 − 0.726i)3-s + (0.618 + 1.90i)5-s + (0.809 + 0.587i)7-s + (−0.454 + 1.40i)9-s + (3.30 + 0.224i)11-s + (−0.381 + 1.17i)13-s + (2 + 1.45i)15-s + (−1.23 − 3.80i)17-s + (1.61 − 1.17i)19-s + 1.23·21-s − 1.61·23-s + (0.809 − 0.587i)25-s + (1.70 + 5.25i)27-s + (−4.73 − 3.44i)29-s + (1.38 − 4.25i)31-s + ⋯
L(s)  = 1  + (0.577 − 0.419i)3-s + (0.276 + 0.850i)5-s + (0.305 + 0.222i)7-s + (−0.151 + 0.466i)9-s + (0.997 + 0.0676i)11-s + (−0.105 + 0.326i)13-s + (0.516 + 0.375i)15-s + (−0.299 − 0.922i)17-s + (0.371 − 0.269i)19-s + 0.269·21-s − 0.337·23-s + (0.161 − 0.117i)25-s + (0.328 + 1.01i)27-s + (−0.879 − 0.638i)29-s + (0.248 − 0.763i)31-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((0.957−0.288i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((0.957−0.288i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: 0.957−0.288i0.957 - 0.288i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(141,⋅)\chi_{308} (141, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), 0.957−0.288i)(2,\ 308,\ (\ :1/2),\ 0.957 - 0.288i)

Particular Values

L(1)L(1) ≈\approx 1.65691+0.243789i1.65691 + 0.243789i
L(12)L(\frac12) ≈\approx 1.65691+0.243789i1.65691 + 0.243789i
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
7 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
11 1+(−3.30−0.224i)T 1 + (-3.30 - 0.224i)T
good3 1+(−1+0.726i)T+(0.927−2.85i)T2 1 + (-1 + 0.726i)T + (0.927 - 2.85i)T^{2}
5 1+(−0.618−1.90i)T+(−4.04+2.93i)T2 1 + (-0.618 - 1.90i)T + (-4.04 + 2.93i)T^{2}
13 1+(0.381−1.17i)T+(−10.5−7.64i)T2 1 + (0.381 - 1.17i)T + (-10.5 - 7.64i)T^{2}
17 1+(1.23+3.80i)T+(−13.7+9.99i)T2 1 + (1.23 + 3.80i)T + (-13.7 + 9.99i)T^{2}
19 1+(−1.61+1.17i)T+(5.87−18.0i)T2 1 + (-1.61 + 1.17i)T + (5.87 - 18.0i)T^{2}
23 1+1.61T+23T2 1 + 1.61T + 23T^{2}
29 1+(4.73+3.44i)T+(8.96+27.5i)T2 1 + (4.73 + 3.44i)T + (8.96 + 27.5i)T^{2}
31 1+(−1.38+4.25i)T+(−25.0−18.2i)T2 1 + (-1.38 + 4.25i)T + (-25.0 - 18.2i)T^{2}
37 1+(2.54+1.84i)T+(11.4+35.1i)T2 1 + (2.54 + 1.84i)T + (11.4 + 35.1i)T^{2}
41 1+(−1.38+1.00i)T+(12.6−38.9i)T2 1 + (-1.38 + 1.00i)T + (12.6 - 38.9i)T^{2}
43 1−4.14T+43T2 1 - 4.14T + 43T^{2}
47 1+(8.09−5.87i)T+(14.5−44.6i)T2 1 + (8.09 - 5.87i)T + (14.5 - 44.6i)T^{2}
53 1+(2.28−7.02i)T+(−42.8−31.1i)T2 1 + (2.28 - 7.02i)T + (-42.8 - 31.1i)T^{2}
59 1+(7.85+5.70i)T+(18.2+56.1i)T2 1 + (7.85 + 5.70i)T + (18.2 + 56.1i)T^{2}
61 1+(2.61+8.05i)T+(−49.3+35.8i)T2 1 + (2.61 + 8.05i)T + (-49.3 + 35.8i)T^{2}
67 1−2.61T+67T2 1 - 2.61T + 67T^{2}
71 1+(−4.04−12.4i)T+(−57.4+41.7i)T2 1 + (-4.04 - 12.4i)T + (-57.4 + 41.7i)T^{2}
73 1+(13.4+9.78i)T+(22.5+69.4i)T2 1 + (13.4 + 9.78i)T + (22.5 + 69.4i)T^{2}
79 1+(−1.57+4.84i)T+(−63.9−46.4i)T2 1 + (-1.57 + 4.84i)T + (-63.9 - 46.4i)T^{2}
83 1+(0.909+2.80i)T+(−67.1+48.7i)T2 1 + (0.909 + 2.80i)T + (-67.1 + 48.7i)T^{2}
89 1+11.2T+89T2 1 + 11.2T + 89T^{2}
97 1+(−3.52+10.8i)T+(−78.4−57.0i)T2 1 + (-3.52 + 10.8i)T + (-78.4 - 57.0i)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

−11.57087416919543000906938075084, −11.00390761598988262054320470850, −9.716103549159145834698239182997, −8.963458669469558636055692239660, −7.79331464307134793872103870826, −7.02246682617515491617292738313, −5.98788465756155210273895720613, −4.56036230669541510202986083107, −3.02928327337187608891074917816, −1.95758593522394858933337205680, 1.46776686721088309497665475727, 3.38074203628068447641306600602, 4.36925146203061652264312814513, 5.59231429805122616705829787919, 6.76037859666680057048885816281, 8.162945899058982246129766692752, 8.884921664889740432602137449633, 9.568609284582158015878531893659, 10.59540264963289674413821966428, 11.76403075056413215079746043261

Graph of the ZZ-function along the critical line