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Properties

Label 2-261-29.28-c1-0-4
Degree 22
Conductor 261261
Sign −i-i
Analytic cond. 2.084092.08409
Root an. cond. 1.443631.44363
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  + 1.66i·2-s − 0.756·4-s + 5.20·7-s + 2.06i·8-s − 4.92i·11-s − 2.72·13-s + 8.64i·14-s − 4.94·16-s + 8.24i·17-s + 8.17·22-s − 5·25-s − 4.52i·26-s − 3.94·28-s − 5.38i·29-s − 4.07i·32-s + ⋯
L(s)  = 1  + 1.17i·2-s − 0.378·4-s + 1.96·7-s + 0.729i·8-s − 1.48i·11-s − 0.755·13-s + 2.31i·14-s − 1.23·16-s + 1.99i·17-s + 1.74·22-s − 25-s − 0.886i·26-s − 0.744·28-s − 0.999i·29-s − 0.720i·32-s + ⋯

Functional equation

Λ(s)=(261s/2 ΓC(s) L(s)=(−i Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(261s/2 ΓC(s+1/2) L(s)=(−i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 261261    =    32⋅293^{2} \cdot 29
Sign: −i-i
Analytic conductor: 2.084092.08409
Root analytic conductor: 1.443631.44363
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ261(28,⋅)\chi_{261} (28, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 261, ( :1/2), −i)(2,\ 261,\ (\ :1/2),\ -i)

Particular Values

L(1)L(1) ≈\approx 1.07484+1.07484i1.07484 + 1.07484i
L(12)L(\frac12) ≈\approx 1.07484+1.07484i1.07484 + 1.07484i
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 1
29 1+5.38iT 1 + 5.38iT
good2 1−1.66iT−2T2 1 - 1.66iT - 2T^{2}
5 1+5T2 1 + 5T^{2}
7 1−5.20T+7T2 1 - 5.20T + 7T^{2}
11 1+4.92iT−11T2 1 + 4.92iT - 11T^{2}
13 1+2.72T+13T2 1 + 2.72T + 13T^{2}
17 1−8.24iT−17T2 1 - 8.24iT - 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1+23T2 1 + 23T^{2}
31 1−31T2 1 - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1+10.7iT−41T2 1 + 10.7iT - 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+1.71iT−47T2 1 + 1.71iT - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1+10.6T+67T2 1 + 10.6T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1−5.03iT−89T2 1 - 5.03iT - 89T^{2}
97 1−97T2 1 - 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

−12.03652163979263385801882432372, −11.24822468560849133378808567378, −10.52289335232629618531489032354, −8.726773607157863540082459929124, −8.176521000502004765071255331715, −7.51309383531782208911058307717, −6.06170509694916636370863239081, −5.40524839820306090859703475933, −4.15989143815161087958658312826, −1.98572221001153328727913236702, 1.56883344954744370225912333409, 2.59583149722910636997990759134, 4.44485243912479532962472500168, 5.03589232272127142242340766968, 7.09906025136613125177566008306, 7.74572370966521745757064712326, 9.234066192340207170951113841802, 9.998474436792621228958608904959, 11.01758936567099582910865255609, 11.77960916034356913783494753256

Graph of the ZZ-function along the critical line