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Properties

Label 2-975-13.12-c1-0-27
Degree 22
Conductor 975975
Sign 0.712−0.701i0.712 - 0.701i
Analytic cond. 7.785417.78541
Root an. cond. 2.790232.79023
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  + 0.490i·2-s + 3-s + 1.75·4-s + 0.490i·6-s + 3.59i·7-s + 1.84i·8-s + 9-s − 4.35i·11-s + 1.75·12-s + (2.56 − 2.53i)13-s − 1.75·14-s + 2.61·16-s + 4.13·17-s + 0.490i·18-s − 1.74i·19-s + ⋯
L(s)  = 1  + 0.346i·2-s + 0.577·3-s + 0.879·4-s + 0.200i·6-s + 1.35i·7-s + 0.651i·8-s + 0.333·9-s − 1.31i·11-s + 0.508·12-s + (0.712 − 0.701i)13-s − 0.470·14-s + 0.654·16-s + 1.00·17-s + 0.115i·18-s − 0.401i·19-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((0.712−0.701i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s+1/2) L(s)=((0.712−0.701i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: 0.712−0.701i0.712 - 0.701i
Analytic conductor: 7.785417.78541
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ975(376,⋅)\chi_{975} (376, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :1/2), 0.712−0.701i)(2,\ 975,\ (\ :1/2),\ 0.712 - 0.701i)

Particular Values

L(1)L(1) ≈\approx 2.35513+0.965376i2.35513 + 0.965376i
L(12)L(\frac12) ≈\approx 2.35513+0.965376i2.35513 + 0.965376i
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)
bad3 1−T 1 - T
5 1 1
13 1+(−2.56+2.53i)T 1 + (-2.56 + 2.53i)T
good2 1−0.490iT−2T2 1 - 0.490iT - 2T^{2}
7 1−3.59iT−7T2 1 - 3.59iT - 7T^{2}
11 1+4.35iT−11T2 1 + 4.35iT - 11T^{2}
17 1−4.13T+17T2 1 - 4.13T + 17T^{2}
19 1+1.74iT−19T2 1 + 1.74iT - 19T^{2}
23 1+0.376T+23T2 1 + 0.376T + 23T^{2}
29 1+7.65T+29T2 1 + 7.65T + 29T^{2}
31 1−6.90iT−31T2 1 - 6.90iT - 31T^{2}
37 1−4.84iT−37T2 1 - 4.84iT - 37T^{2}
41 1−4.84iT−41T2 1 - 4.84iT - 41T^{2}
43 1+3.24T+43T2 1 + 3.24T + 43T^{2}
47 1−6.13iT−47T2 1 - 6.13iT - 47T^{2}
53 1+3.75T+53T2 1 + 3.75T + 53T^{2}
59 1+13.8iT−59T2 1 + 13.8iT - 59T^{2}
61 1−8.51T+61T2 1 - 8.51T + 61T^{2}
67 1+1.25iT−67T2 1 + 1.25iT - 67T^{2}
71 1+11.4iT−71T2 1 + 11.4iT - 71T^{2}
73 1−13.5iT−73T2 1 - 13.5iT - 73T^{2}
79 1+9.41T+79T2 1 + 9.41T + 79T^{2}
83 1+16.8iT−83T2 1 + 16.8iT - 83T^{2}
89 1+2.94iT−89T2 1 + 2.94iT - 89T^{2}
97 1+10.2iT−97T2 1 + 10.2iT - 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

−10.06719352700627594497283171481, −9.026728205487613830573878401337, −8.333444715988221221367529590122, −7.82904568684072569157790267318, −6.61246891754751410681127130749, −5.83756319945862398372874894721, −5.25706363320167939628122922787, −3.34814669084853667083681840853, −2.90359217461147875519540478929, −1.55559296063227462788223600377, 1.32274828871299781872996439143, 2.25280619890136800859266393044, 3.71794194310207854824879810940, 4.08203230302237966033542224585, 5.64588291446288888320048038868, 6.85714684729863027113627167539, 7.32733779685773957088343601979, 7.991449125733700015823269466319, 9.347554166000611289473096622580, 10.00239102394222759187275905225

Graph of the ZZ-function along the critical line