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Properties

Label 2-975-13.12-c1-0-42
Degree 22
Conductor 975975
Sign 0.554+0.832i0.554 + 0.832i
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  + 3-s + 2·4-s − 3i·7-s + 9-s − 3i·11-s + 2·12-s + (−2 − 3i)13-s + 4·16-s − 3·17-s − 3i·21-s + 3·23-s + 27-s − 6i·28-s − 6·29-s − 6i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 4-s − 1.13i·7-s + 0.333·9-s − 0.904i·11-s + 0.577·12-s + (−0.554 − 0.832i)13-s + 16-s − 0.727·17-s − 0.654i·21-s + 0.625·23-s + 0.192·27-s − 1.13i·28-s − 1.11·29-s − 1.07i·31-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((0.554+0.832i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s+1/2) L(s)=((0.554+0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: 0.554+0.832i0.554 + 0.832i
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), 0.554+0.832i)(2,\ 975,\ (\ :1/2),\ 0.554 + 0.832i)

Particular Values

L(1)L(1) ≈\approx 2.07286−1.10936i2.07286 - 1.10936i
L(12)L(\frac12) ≈\approx 2.07286−1.10936i2.07286 - 1.10936i
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+3i)T 1 + (2 + 3i)T
good2 1−2T2 1 - 2T^{2}
7 1+3iT−7T2 1 + 3iT - 7T^{2}
11 1+3iT−11T2 1 + 3iT - 11T^{2}
17 1+3T+17T2 1 + 3T + 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1−3T+23T2 1 - 3T + 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1+6iT−31T2 1 + 6iT - 31T^{2}
37 1−9iT−37T2 1 - 9iT - 37T^{2}
41 1−3iT−41T2 1 - 3iT - 41T^{2}
43 1−10T+43T2 1 - 10T + 43T^{2}
47 1−12iT−47T2 1 - 12iT - 47T^{2}
53 1−3T+53T2 1 - 3T + 53T^{2}
59 1−12iT−59T2 1 - 12iT - 59T^{2}
61 1−T+61T2 1 - T + 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+9iT−71T2 1 + 9iT - 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1−T+79T2 1 - T + 79T^{2}
83 1−6iT−83T2 1 - 6iT - 83T^{2}
89 1−15iT−89T2 1 - 15iT - 89T^{2}
97 1−9iT−97T2 1 - 9iT - 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

−9.983133119649501121485329400809, −9.062140486490919815802981401847, −7.892480404282757024996330305589, −7.53542134176631198358354871038, −6.60410925044539942272181830883, −5.71115413057068205718745805993, −4.38935995798510647778251278164, −3.34249178551493308952259467660, −2.50463904316358377716741903455, −1.02031043262301699410085471867, 1.98986558693205345866975887997, 2.37996971644120842624565130602, 3.69193104071670533041584123181, 4.93994687541428352581621825825, 5.90701223857490585696535836757, 7.03093684242178944687350142204, 7.34594096643427722638850360868, 8.651257462237378221770982044723, 9.181199570012231025496148979138, 10.06065802446174501461455096350

Graph of the ZZ-function along the critical line