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Properties

Label 2-1530-5.4-c1-0-32
Degree 22
Conductor 15301530
Sign ii
Analytic cond. 12.217112.2171
Root an. cond. 3.495293.49529
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  + i·2-s − 4-s + 2.23i·5-s − 3.23i·7-s − i·8-s − 2.23·10-s − 2·11-s + 0.763i·13-s + 3.23·14-s + 16-s + i·17-s − 2.47·19-s − 2.23i·20-s − 2i·22-s − 4i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.999i·5-s − 1.22i·7-s − 0.353i·8-s − 0.707·10-s − 0.603·11-s + 0.211i·13-s + 0.864·14-s + 0.250·16-s + 0.242i·17-s − 0.567·19-s − 0.499i·20-s − 0.426i·22-s − 0.834i·23-s + ⋯

Functional equation

Λ(s)=(1530s/2 ΓC(s) L(s)=(i Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1530s/2 ΓC(s+1/2) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15301530    =    2⋅32⋅5⋅172 \cdot 3^{2} \cdot 5 \cdot 17
Sign: ii
Analytic conductor: 12.217112.2171
Root analytic conductor: 3.495293.49529
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1530(919,⋅)\chi_{1530} (919, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1530, ( :1/2), i)(2,\ 1530,\ (\ :1/2),\ i)

Particular Values

L(1)L(1) ≈\approx 0.39922529650.3992252965
L(12)L(\frac12) ≈\approx 0.39922529650.3992252965
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−iT 1 - iT
3 1 1
5 1−2.23iT 1 - 2.23iT
17 1−iT 1 - iT
good7 1+3.23iT−7T2 1 + 3.23iT - 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1−0.763iT−13T2 1 - 0.763iT - 13T^{2}
19 1+2.47T+19T2 1 + 2.47T + 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
29 1+4T+29T2 1 + 4T + 29T^{2}
31 1+6.47T+31T2 1 + 6.47T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+5.70T+41T2 1 + 5.70T + 41T^{2}
43 1+10.1iT−43T2 1 + 10.1iT - 43T^{2}
47 1−1.52iT−47T2 1 - 1.52iT - 47T^{2}
53 1+6.94iT−53T2 1 + 6.94iT - 53T^{2}
59 1+1.70T+59T2 1 + 1.70T + 59T^{2}
61 1+4.47T+61T2 1 + 4.47T + 61T^{2}
67 1+11.7iT−67T2 1 + 11.7iT - 67T^{2}
71 1+6.76T+71T2 1 + 6.76T + 71T^{2}
73 1−13.2iT−73T2 1 - 13.2iT - 73T^{2}
79 1−16.9T+79T2 1 - 16.9T + 79T^{2}
83 1+1.52iT−83T2 1 + 1.52iT - 83T^{2}
89 1−3.52T+89T2 1 - 3.52T + 89T^{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

−9.210854202503868203551152554487, −8.233093154233786589490847563469, −7.45256697054074903015477362962, −6.95612248995389147347192175370, −6.20874856829200788660934405213, −5.20865955361717963550896491072, −4.12332053070412539081218670458, −3.44729324900800156391099060520, −2.05322668054936673380114053844, −0.15128576267795183587833397861, 1.53227414844028222275862722486, 2.49479410890826440591981969838, 3.57879529500617047421428488749, 4.73657189795930188027600953159, 5.38553090601852159792908910602, 6.07082410406131634535431622837, 7.55452545554919402999044052426, 8.274554493553103265466153121494, 9.073191396631734250917509722633, 9.444485548926707973619445412577

Graph of the ZZ-function along the critical line