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Properties

Label 2-1176-8.5-c1-0-21
Degree 22
Conductor 11761176
Sign −0.726−0.687i-0.726 - 0.687i
Analytic cond. 9.390409.39040
Root an. cond. 3.064373.06437
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  + (−0.353 + 1.36i)2-s + i·3-s + (−1.75 − 0.967i)4-s − 0.677i·5-s + (−1.36 − 0.353i)6-s + (1.94 − 2.05i)8-s − 9-s + (0.927 + 0.239i)10-s + 1.67i·11-s + (0.967 − 1.75i)12-s + 1.28i·13-s + 0.677·15-s + (2.12 + 3.38i)16-s + 6.36·17-s + (0.353 − 1.36i)18-s − 2.55i·19-s + ⋯
L(s)  = 1  + (−0.249 + 0.968i)2-s + 0.577i·3-s + (−0.875 − 0.483i)4-s − 0.302i·5-s + (−0.559 − 0.144i)6-s + (0.687 − 0.726i)8-s − 0.333·9-s + (0.293 + 0.0756i)10-s + 0.503i·11-s + (0.279 − 0.505i)12-s + 0.355i·13-s + 0.174·15-s + (0.531 + 0.846i)16-s + 1.54·17-s + (0.0832 − 0.322i)18-s − 0.585i·19-s + ⋯

Functional equation

Λ(s)=(1176s/2 ΓC(s) L(s)=((−0.726−0.687i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1176s/2 ΓC(s+1/2) L(s)=((−0.726−0.687i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11761176    =    23⋅3⋅722^{3} \cdot 3 \cdot 7^{2}
Sign: −0.726−0.687i-0.726 - 0.687i
Analytic conductor: 9.390409.39040
Root analytic conductor: 3.064373.06437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1176(589,⋅)\chi_{1176} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1176, ( :1/2), −0.726−0.687i)(2,\ 1176,\ (\ :1/2),\ -0.726 - 0.687i)

Particular Values

L(1)L(1) ≈\approx 1.1984332941.198433294
L(12)L(\frac12) ≈\approx 1.1984332941.198433294
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+(0.353−1.36i)T 1 + (0.353 - 1.36i)T
3 1−iT 1 - iT
7 1 1
good5 1+0.677iT−5T2 1 + 0.677iT - 5T^{2}
11 1−1.67iT−11T2 1 - 1.67iT - 11T^{2}
13 1−1.28iT−13T2 1 - 1.28iT - 13T^{2}
17 1−6.36T+17T2 1 - 6.36T + 17T^{2}
19 1+2.55iT−19T2 1 + 2.55iT - 19T^{2}
23 1+0.255T+23T2 1 + 0.255T + 23T^{2}
29 1−6.27iT−29T2 1 - 6.27iT - 29T^{2}
31 1+4.28T+31T2 1 + 4.28T + 31T^{2}
37 1−6.49iT−37T2 1 - 6.49iT - 37T^{2}
41 1+6.43T+41T2 1 + 6.43T + 41T^{2}
43 1−5.48iT−43T2 1 - 5.48iT - 43T^{2}
47 1−9.46T+47T2 1 - 9.46T + 47T^{2}
53 1−5.67iT−53T2 1 - 5.67iT - 53T^{2}
59 1+10.0iT−59T2 1 + 10.0iT - 59T^{2}
61 1−15.2iT−61T2 1 - 15.2iT - 61T^{2}
67 1−15.7iT−67T2 1 - 15.7iT - 67T^{2}
71 1+5.48T+71T2 1 + 5.48T + 71T^{2}
73 1−2.86T+73T2 1 - 2.86T + 73T^{2}
79 1+12.1T+79T2 1 + 12.1T + 79T^{2}
83 1+7.63iT−83T2 1 + 7.63iT - 83T^{2}
89 1−6.80T+89T2 1 - 6.80T + 89T^{2}
97 1+0.477T+97T2 1 + 0.477T + 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

−9.982311405719972600910347197315, −9.076918152685356013062304276698, −8.594059998589453518337370968109, −7.53954847827431288959276326665, −6.89868838569505661799336154211, −5.78295649378656043678345634263, −5.06798423039091446350988734288, −4.31815485731476592542603498027, −3.16338950347352443313597429646, −1.25445151231524135037595649461, 0.66254226534169940124304381502, 1.93774902825428083885418273904, 3.08028734998574170878697989437, 3.80276961577537153861046095092, 5.21044711639607271806394193618, 5.96503618725259311379554252311, 7.26753931807026089272676095138, 7.923769214911342374822370622411, 8.673183888865787602572940412564, 9.591155278346392311773493716358

Graph of the ZZ-function along the critical line