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Properties

Label 2-1170-5.4-c1-0-8
Degree 22
Conductor 11701170
Sign −0.749−0.662i-0.749 - 0.662i
Analytic cond. 9.342499.34249
Root an. cond. 3.056543.05654
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 + (1.67 + 1.48i)5-s − 1.19i·7-s − i·8-s + (−1.48 + 1.67i)10-s − 2·11-s + i·13-s + 1.19·14-s + 16-s + 4.54i·17-s − 4.15·19-s + (−1.67 − 1.48i)20-s − 2i·22-s + 7.11i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.749 + 0.662i)5-s − 0.451i·7-s − 0.353i·8-s + (−0.468 + 0.529i)10-s − 0.603·11-s + 0.277i·13-s + 0.319·14-s + 0.250·16-s + 1.10i·17-s − 0.953·19-s + (−0.374 − 0.331i)20-s − 0.426i·22-s + 1.48i·23-s + ⋯

Functional equation

Λ(s)=(1170s/2 ΓC(s) L(s)=((−0.749−0.662i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1170s/2 ΓC(s+1/2) L(s)=((−0.749−0.662i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11701170    =    2⋅32⋅5⋅132 \cdot 3^{2} \cdot 5 \cdot 13
Sign: −0.749−0.662i-0.749 - 0.662i
Analytic conductor: 9.342499.34249
Root analytic conductor: 3.056543.05654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1170(469,⋅)\chi_{1170} (469, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1170, ( :1/2), −0.749−0.662i)(2,\ 1170,\ (\ :1/2),\ -0.749 - 0.662i)

Particular Values

L(1)L(1) ≈\approx 1.3851658321.385165832
L(12)L(\frac12) ≈\approx 1.3851658321.385165832
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+(−1.67−1.48i)T 1 + (-1.67 - 1.48i)T
13 1−iT 1 - iT
good7 1+1.19iT−7T2 1 + 1.19iT - 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
17 1−4.54iT−17T2 1 - 4.54iT - 17T^{2}
19 1+4.15T+19T2 1 + 4.15T + 19T^{2}
23 1−7.11iT−23T2 1 - 7.11iT - 23T^{2}
29 1−10.7T+29T2 1 - 10.7T + 29T^{2}
31 1+5.35T+31T2 1 + 5.35T + 31T^{2}
37 1−3.92iT−37T2 1 - 3.92iT - 37T^{2}
41 1+1.03T+41T2 1 + 1.03T + 41T^{2}
43 1−10.8iT−43T2 1 - 10.8iT - 43T^{2}
47 1+1.61iT−47T2 1 + 1.61iT - 47T^{2}
53 1+4.18iT−53T2 1 + 4.18iT - 53T^{2}
59 1+2.31T+59T2 1 + 2.31T + 59T^{2}
61 1+7.08T+61T2 1 + 7.08T + 61T^{2}
67 1−4.70iT−67T2 1 - 4.70iT - 67T^{2}
71 1−9.27T+71T2 1 - 9.27T + 71T^{2}
73 1+3.58iT−73T2 1 + 3.58iT - 73T^{2}
79 1+15.1T+79T2 1 + 15.1T + 79T^{2}
83 1+1.73iT−83T2 1 + 1.73iT - 83T^{2}
89 1−14.3T+89T2 1 - 14.3T + 89T^{2}
97 1+9.19iT−97T2 1 + 9.19iT - 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

−10.15331073648043388827079556744, −9.253465142121131553566013293366, −8.329761195402655914272559429820, −7.55835588383092890203189883191, −6.65740643911797505735500673879, −6.09020036123218662411855144244, −5.15516370501241418437467989610, −4.10164632192913681471296729668, −2.98523700247761674794036627095, −1.61846595725382923159618177305, 0.59569675783565761835236191891, 2.14131810951456483567287302779, 2.81781449182612074086500396360, 4.33768758611873188842958745859, 5.06835332678259619329547873917, 5.87161091555660559148178774033, 6.90431656660370579667247754443, 8.199298993207656235020206543649, 8.782518858080373928462867965156, 9.460903939366425349065129410599

Graph of the ZZ-function along the critical line