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Properties

Label 2-304-16.13-c1-0-1
Degree 22
Conductor 304304
Sign 0.982−0.187i0.982 - 0.187i
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  + (−0.724 − 1.21i)2-s + (−2.33 − 2.33i)3-s + (−0.949 + 1.76i)4-s + (0.751 − 0.751i)5-s + (−1.14 + 4.53i)6-s + 3.10i·7-s + (2.82 − 0.122i)8-s + 7.92i·9-s + (−1.45 − 0.368i)10-s + (−2.88 + 2.88i)11-s + (6.33 − 1.89i)12-s + (−1.54 − 1.54i)13-s + (3.77 − 2.25i)14-s − 3.51·15-s + (−2.19 − 3.34i)16-s + 3.53·17-s + ⋯
L(s)  = 1  + (−0.512 − 0.858i)2-s + (−1.34 − 1.34i)3-s + (−0.474 + 0.880i)4-s + (0.336 − 0.336i)5-s + (−0.467 + 1.85i)6-s + 1.17i·7-s + (0.999 − 0.0433i)8-s + 2.64i·9-s + (−0.461 − 0.116i)10-s + (−0.869 + 0.869i)11-s + (1.82 − 0.547i)12-s + (−0.427 − 0.427i)13-s + (1.00 − 0.601i)14-s − 0.907·15-s + (−0.549 − 0.835i)16-s + 0.857·17-s + ⋯

Functional equation

Λ(s)=(304s/2 ΓC(s) L(s)=((0.982−0.187i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(304s/2 ΓC(s+1/2) L(s)=((0.982−0.187i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 304304    =    24⋅192^{4} \cdot 19
Sign: 0.982−0.187i0.982 - 0.187i
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), 0.982−0.187i)(2,\ 304,\ (\ :1/2),\ 0.982 - 0.187i)

Particular Values

L(1)L(1) ≈\approx 0.378952+0.0358634i0.378952 + 0.0358634i
L(12)L(\frac12) ≈\approx 0.378952+0.0358634i0.378952 + 0.0358634i
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+(0.724+1.21i)T 1 + (0.724 + 1.21i)T
19 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good3 1+(2.33+2.33i)T+3iT2 1 + (2.33 + 2.33i)T + 3iT^{2}
5 1+(−0.751+0.751i)T−5iT2 1 + (-0.751 + 0.751i)T - 5iT^{2}
7 1−3.10iT−7T2 1 - 3.10iT - 7T^{2}
11 1+(2.88−2.88i)T−11iT2 1 + (2.88 - 2.88i)T - 11iT^{2}
13 1+(1.54+1.54i)T+13iT2 1 + (1.54 + 1.54i)T + 13iT^{2}
17 1−3.53T+17T2 1 - 3.53T + 17T^{2}
23 1−1.42iT−23T2 1 - 1.42iT - 23T^{2}
29 1+(−4.10−4.10i)T+29iT2 1 + (-4.10 - 4.10i)T + 29iT^{2}
31 1+8.86T+31T2 1 + 8.86T + 31T^{2}
37 1+(5.87−5.87i)T−37iT2 1 + (5.87 - 5.87i)T - 37iT^{2}
41 1+1.90iT−41T2 1 + 1.90iT - 41T^{2}
43 1+(−1.59+1.59i)T−43iT2 1 + (-1.59 + 1.59i)T - 43iT^{2}
47 1−9.75T+47T2 1 - 9.75T + 47T^{2}
53 1+(4.03−4.03i)T−53iT2 1 + (4.03 - 4.03i)T - 53iT^{2}
59 1+(7.96−7.96i)T−59iT2 1 + (7.96 - 7.96i)T - 59iT^{2}
61 1+(2.91+2.91i)T+61iT2 1 + (2.91 + 2.91i)T + 61iT^{2}
67 1+(−7.69−7.69i)T+67iT2 1 + (-7.69 - 7.69i)T + 67iT^{2}
71 1−7.79iT−71T2 1 - 7.79iT - 71T^{2}
73 1+3.45iT−73T2 1 + 3.45iT - 73T^{2}
79 1+0.238T+79T2 1 + 0.238T + 79T^{2}
83 1+(−6.68−6.68i)T+83iT2 1 + (-6.68 - 6.68i)T + 83iT^{2}
89 1+0.325iT−89T2 1 + 0.325iT - 89T^{2}
97 1+9.17T+97T2 1 + 9.17T + 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

−12.00325190874043249293397585514, −10.94783602074138175186747605472, −10.17316653828883723837794152551, −8.960431975751196956389095026547, −7.80990744489252518529724286073, −7.10529342225056890617717156970, −5.55665523350332250548651284668, −5.08639023503350074968099012887, −2.57783154535286735552438969150, −1.50325409218188584037416731360, 0.40436506226383474764134559117, 3.81213803040901964678484342592, 4.87466336927626937547616808882, 5.75660423867300569589325231038, 6.57611945173323588707534101196, 7.71009662279197562175683391778, 9.126398600702232739514827455780, 10.06630185629448258216462025274, 10.54749796693590551749957911518, 11.08995126689734953194949043078

Graph of the ZZ-function along the critical line