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Properties

Label 2-308-28.19-c1-0-19
Degree 22
Conductor 308308
Sign −0.237+0.971i-0.237 + 0.971i
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  + (−0.647 − 1.25i)2-s + (−0.656 − 1.13i)3-s + (−1.16 + 1.62i)4-s + (0.985 + 0.569i)5-s + (−1.00 + 1.56i)6-s + (2.64 + 0.0442i)7-s + (2.79 + 0.407i)8-s + (0.639 − 1.10i)9-s + (0.0776 − 1.60i)10-s + (0.866 − 0.5i)11-s + (2.61 + 0.252i)12-s − 0.913i·13-s + (−1.65 − 3.35i)14-s − 1.49i·15-s + (−1.29 − 3.78i)16-s + (−2.27 + 1.31i)17-s + ⋯
L(s)  = 1  + (−0.457 − 0.889i)2-s + (−0.378 − 0.656i)3-s + (−0.581 + 0.813i)4-s + (0.440 + 0.254i)5-s + (−0.409 + 0.637i)6-s + (0.999 + 0.0167i)7-s + (0.989 + 0.144i)8-s + (0.213 − 0.368i)9-s + (0.0245 − 0.508i)10-s + (0.261 − 0.150i)11-s + (0.754 + 0.0729i)12-s − 0.253i·13-s + (−0.442 − 0.896i)14-s − 0.385i·15-s + (−0.324 − 0.945i)16-s + (−0.550 + 0.317i)17-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((−0.237+0.971i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((−0.237+0.971i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: −0.237+0.971i-0.237 + 0.971i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(243,⋅)\chi_{308} (243, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), −0.237+0.971i)(2,\ 308,\ (\ :1/2),\ -0.237 + 0.971i)

Particular Values

L(1)L(1) ≈\approx 0.647680−0.825268i0.647680 - 0.825268i
L(12)L(\frac12) ≈\approx 0.647680−0.825268i0.647680 - 0.825268i
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+(0.647+1.25i)T 1 + (0.647 + 1.25i)T
7 1+(−2.64−0.0442i)T 1 + (-2.64 - 0.0442i)T
11 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
good3 1+(0.656+1.13i)T+(−1.5+2.59i)T2 1 + (0.656 + 1.13i)T + (-1.5 + 2.59i)T^{2}
5 1+(−0.985−0.569i)T+(2.5+4.33i)T2 1 + (-0.985 - 0.569i)T + (2.5 + 4.33i)T^{2}
13 1+0.913iT−13T2 1 + 0.913iT - 13T^{2}
17 1+(2.27−1.31i)T+(8.5−14.7i)T2 1 + (2.27 - 1.31i)T + (8.5 - 14.7i)T^{2}
19 1+(−2.21+3.83i)T+(−9.5−16.4i)T2 1 + (-2.21 + 3.83i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.84−1.64i)T+(11.5+19.9i)T2 1 + (-2.84 - 1.64i)T + (11.5 + 19.9i)T^{2}
29 1+1.58T+29T2 1 + 1.58T + 29T^{2}
31 1+(−1.70−2.95i)T+(−15.5+26.8i)T2 1 + (-1.70 - 2.95i)T + (-15.5 + 26.8i)T^{2}
37 1+(−3.33+5.77i)T+(−18.5−32.0i)T2 1 + (-3.33 + 5.77i)T + (-18.5 - 32.0i)T^{2}
41 1−1.54iT−41T2 1 - 1.54iT - 41T^{2}
43 1+2.10iT−43T2 1 + 2.10iT - 43T^{2}
47 1+(−1.31+2.26i)T+(−23.5−40.7i)T2 1 + (-1.31 + 2.26i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.812−1.40i)T+(−26.5+45.8i)T2 1 + (-0.812 - 1.40i)T + (-26.5 + 45.8i)T^{2}
59 1+(−0.816−1.41i)T+(−29.5+51.0i)T2 1 + (-0.816 - 1.41i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.716+0.413i)T+(30.5+52.8i)T2 1 + (0.716 + 0.413i)T + (30.5 + 52.8i)T^{2}
67 1+(−7.82+4.51i)T+(33.5−58.0i)T2 1 + (-7.82 + 4.51i)T + (33.5 - 58.0i)T^{2}
71 1−12.4iT−71T2 1 - 12.4iT - 71T^{2}
73 1+(9.92−5.72i)T+(36.5−63.2i)T2 1 + (9.92 - 5.72i)T + (36.5 - 63.2i)T^{2}
79 1+(−8.05−4.64i)T+(39.5+68.4i)T2 1 + (-8.05 - 4.64i)T + (39.5 + 68.4i)T^{2}
83 1+15.4T+83T2 1 + 15.4T + 83T^{2}
89 1+(−14.0−8.13i)T+(44.5+77.0i)T2 1 + (-14.0 - 8.13i)T + (44.5 + 77.0i)T^{2}
97 1−11.7iT−97T2 1 - 11.7iT - 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

−11.40902890592369240389462993199, −10.72701297647251105076388555120, −9.616900328520744588764599390689, −8.732298027880279009990275941674, −7.66258835868637713809269187496, −6.73419291743273993673403494103, −5.34106902467716146568087316497, −4.02165994996380940765854771292, −2.40544280688134646710649393864, −1.11078676333822495356327542555, 1.65145875527739229137556352988, 4.29635646609772477604239495251, 5.04552391379963822253811173291, 5.93358763161503586233661472945, 7.24928151668202434314886768896, 8.128377067565997527919309055392, 9.195337613636126964062936116749, 9.925608840015060366410982622569, 10.84437462833156606669799796741, 11.64654199113882629234101928875

Graph of the ZZ-function along the critical line