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Properties

Label 2-7581-1.1-c1-0-84
Degree 22
Conductor 75817581
Sign 11
Analytic cond. 60.534560.5345
Root an. cond. 7.780397.78039
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.78·2-s + 3-s + 5.78·4-s − 0.430·5-s − 2.78·6-s + 7-s − 10.5·8-s + 9-s + 1.20·10-s − 4.67·11-s + 5.78·12-s + 4.61·13-s − 2.78·14-s − 0.430·15-s + 17.8·16-s + 5.52·17-s − 2.78·18-s − 2.49·20-s + 21-s + 13.0·22-s − 1.98·23-s − 10.5·24-s − 4.81·25-s − 12.8·26-s + 27-s + 5.78·28-s − 6.41·29-s + ⋯
L(s)  = 1  − 1.97·2-s + 0.577·3-s + 2.89·4-s − 0.192·5-s − 1.13·6-s + 0.377·7-s − 3.72·8-s + 0.333·9-s + 0.380·10-s − 1.40·11-s + 1.66·12-s + 1.27·13-s − 0.745·14-s − 0.111·15-s + 4.46·16-s + 1.33·17-s − 0.657·18-s − 0.557·20-s + 0.218·21-s + 2.77·22-s − 0.413·23-s − 2.15·24-s − 0.962·25-s − 2.52·26-s + 0.192·27-s + 1.09·28-s − 1.19·29-s + ⋯

Functional equation

Λ(s)=(7581s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7581 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7581s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7581 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 75817581    =    3⋅7⋅1923 \cdot 7 \cdot 19^{2}
Sign: 11
Analytic conductor: 60.534560.5345
Root analytic conductor: 7.780397.78039
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7581, ( :1/2), 1)(2,\ 7581,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.98820110460.9882011046
L(12)L(\frac12) ≈\approx 0.98820110460.9882011046
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
7 1−T 1 - T
19 1 1
good2 1+2.78T+2T2 1 + 2.78T + 2T^{2}
5 1+0.430T+5T2 1 + 0.430T + 5T^{2}
11 1+4.67T+11T2 1 + 4.67T + 11T^{2}
13 1−4.61T+13T2 1 - 4.61T + 13T^{2}
17 1−5.52T+17T2 1 - 5.52T + 17T^{2}
23 1+1.98T+23T2 1 + 1.98T + 23T^{2}
29 1+6.41T+29T2 1 + 6.41T + 29T^{2}
31 1−6.95T+31T2 1 - 6.95T + 31T^{2}
37 1−3.95T+37T2 1 - 3.95T + 37T^{2}
41 1−0.487T+41T2 1 - 0.487T + 41T^{2}
43 1+2.25T+43T2 1 + 2.25T + 43T^{2}
47 1−7.10T+47T2 1 - 7.10T + 47T^{2}
53 1−7.50T+53T2 1 - 7.50T + 53T^{2}
59 1−2.68T+59T2 1 - 2.68T + 59T^{2}
61 1+2.22T+61T2 1 + 2.22T + 61T^{2}
67 1+0.358T+67T2 1 + 0.358T + 67T^{2}
71 1−13.0T+71T2 1 - 13.0T + 71T^{2}
73 1+1.77T+73T2 1 + 1.77T + 73T^{2}
79 1+13.1T+79T2 1 + 13.1T + 79T^{2}
83 1−2.40T+83T2 1 - 2.40T + 83T^{2}
89 1+16.4T+89T2 1 + 16.4T + 89T^{2}
97 1+3.22T+97T2 1 + 3.22T + 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

−8.023658275420515524088820610319, −7.64760624976741389103136464849, −6.92920499067228837633941384469, −5.90829601870201104322759564402, −5.53430529683176933979843754891, −3.97263465568242541579291356637, −3.13231418226514475838773813202, −2.40433078652648909510496114157, −1.57286477329095711094179835687, −0.66937907867223875519700999487, 0.66937907867223875519700999487, 1.57286477329095711094179835687, 2.40433078652648909510496114157, 3.13231418226514475838773813202, 3.97263465568242541579291356637, 5.53430529683176933979843754891, 5.90829601870201104322759564402, 6.92920499067228837633941384469, 7.64760624976741389103136464849, 8.023658275420515524088820610319

Graph of the ZZ-function along the critical line