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Properties

Label 2-308-77.16-c1-0-3
Degree 22
Conductor 308308
Sign 0.752−0.658i0.752 - 0.658i
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  + (−0.0341 + 0.324i)3-s + (2.36 + 2.62i)5-s + (−0.532 − 2.59i)7-s + (2.83 + 0.601i)9-s + (−3.06 + 1.27i)11-s + (0.771 + 2.37i)13-s + (−0.933 + 0.678i)15-s + (−1.54 + 0.327i)17-s + (6.65 − 2.96i)19-s + (0.859 − 0.0846i)21-s + (−3.62 + 6.28i)23-s + (−0.782 + 7.44i)25-s + (−0.594 + 1.83i)27-s + (3.15 − 2.29i)29-s + (3.34 − 3.71i)31-s + ⋯
L(s)  = 1  + (−0.0197 + 0.187i)3-s + (1.05 + 1.17i)5-s + (−0.201 − 0.979i)7-s + (0.943 + 0.200i)9-s + (−0.923 + 0.383i)11-s + (0.213 + 0.658i)13-s + (−0.241 + 0.175i)15-s + (−0.373 + 0.0794i)17-s + (1.52 − 0.679i)19-s + (0.187 − 0.0184i)21-s + (−0.756 + 1.31i)23-s + (−0.156 + 1.48i)25-s + (−0.114 + 0.352i)27-s + (0.585 − 0.425i)29-s + (0.600 − 0.667i)31-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((0.752−0.658i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((0.752−0.658i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: 0.752−0.658i0.752 - 0.658i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(93,⋅)\chi_{308} (93, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), 0.752−0.658i)(2,\ 308,\ (\ :1/2),\ 0.752 - 0.658i)

Particular Values

L(1)L(1) ≈\approx 1.42590+0.535522i1.42590 + 0.535522i
L(12)L(\frac12) ≈\approx 1.42590+0.535522i1.42590 + 0.535522i
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+(0.532+2.59i)T 1 + (0.532 + 2.59i)T
11 1+(3.06−1.27i)T 1 + (3.06 - 1.27i)T
good3 1+(0.0341−0.324i)T+(−2.93−0.623i)T2 1 + (0.0341 - 0.324i)T + (-2.93 - 0.623i)T^{2}
5 1+(−2.36−2.62i)T+(−0.522+4.97i)T2 1 + (-2.36 - 2.62i)T + (-0.522 + 4.97i)T^{2}
13 1+(−0.771−2.37i)T+(−10.5+7.64i)T2 1 + (-0.771 - 2.37i)T + (-10.5 + 7.64i)T^{2}
17 1+(1.54−0.327i)T+(15.5−6.91i)T2 1 + (1.54 - 0.327i)T + (15.5 - 6.91i)T^{2}
19 1+(−6.65+2.96i)T+(12.7−14.1i)T2 1 + (-6.65 + 2.96i)T + (12.7 - 14.1i)T^{2}
23 1+(3.62−6.28i)T+(−11.5−19.9i)T2 1 + (3.62 - 6.28i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3.15+2.29i)T+(8.96−27.5i)T2 1 + (-3.15 + 2.29i)T + (8.96 - 27.5i)T^{2}
31 1+(−3.34+3.71i)T+(−3.24−30.8i)T2 1 + (-3.34 + 3.71i)T + (-3.24 - 30.8i)T^{2}
37 1+(0.638+6.07i)T+(−36.1+7.69i)T2 1 + (0.638 + 6.07i)T + (-36.1 + 7.69i)T^{2}
41 1+(8.07+5.86i)T+(12.6+38.9i)T2 1 + (8.07 + 5.86i)T + (12.6 + 38.9i)T^{2}
43 1−2.83T+43T2 1 - 2.83T + 43T^{2}
47 1+(1.10−0.492i)T+(31.4−34.9i)T2 1 + (1.10 - 0.492i)T + (31.4 - 34.9i)T^{2}
53 1+(1.12−1.25i)T+(−5.54−52.7i)T2 1 + (1.12 - 1.25i)T + (-5.54 - 52.7i)T^{2}
59 1+(4.70+2.09i)T+(39.4+43.8i)T2 1 + (4.70 + 2.09i)T + (39.4 + 43.8i)T^{2}
61 1+(6.27+6.96i)T+(−6.37+60.6i)T2 1 + (6.27 + 6.96i)T + (-6.37 + 60.6i)T^{2}
67 1+(3.03+5.26i)T+(−33.5+58.0i)T2 1 + (3.03 + 5.26i)T + (-33.5 + 58.0i)T^{2}
71 1+(−0.205+0.631i)T+(−57.4−41.7i)T2 1 + (-0.205 + 0.631i)T + (-57.4 - 41.7i)T^{2}
73 1+(−2.52−1.12i)T+(48.8+54.2i)T2 1 + (-2.52 - 1.12i)T + (48.8 + 54.2i)T^{2}
79 1+(15.3+3.27i)T+(72.1+32.1i)T2 1 + (15.3 + 3.27i)T + (72.1 + 32.1i)T^{2}
83 1+(−1.50+4.61i)T+(−67.1−48.7i)T2 1 + (-1.50 + 4.61i)T + (-67.1 - 48.7i)T^{2}
89 1+(−5.86+10.1i)T+(−44.5−77.0i)T2 1 + (-5.86 + 10.1i)T + (-44.5 - 77.0i)T^{2}
97 1+(5.58+17.1i)T+(−78.4+57.0i)T2 1 + (5.58 + 17.1i)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

−11.55985813569956075404833473004, −10.62703193633571565521440823446, −10.01086196329682239724251466486, −9.450865127366265559182441878342, −7.56205945423772668562155904764, −7.08768290436697893998565952303, −5.99438115313032909177491049615, −4.68994648527795703518702040753, −3.34974725248926942998628244288, −1.92101196822121375080855296927, 1.35953071374729915641894913528, 2.85423440748634763554171795830, 4.77947954841924533144085394307, 5.55326153181507717149893638498, 6.46472504492246461335696172230, 8.028986089780057601547154902098, 8.734904814104571410116851676012, 9.795713490592039400931065431726, 10.31508482217885811202487501526, 11.96452729082446397941474354893

Graph of the ZZ-function along the critical line