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Properties

Label 2-334-1.1-c1-0-4
Degree 22
Conductor 334334
Sign 11
Analytic cond. 2.667002.66700
Root an. cond. 1.633091.63309
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-s + 2.82·3-s + 4-s − 0.414·5-s − 2.82·6-s − 3·7-s − 8-s + 5.00·9-s + 0.414·10-s + 2.82·11-s + 2.82·12-s + 6.82·13-s + 3·14-s − 1.17·15-s + 16-s + 4.82·17-s − 5.00·18-s − 3.65·19-s − 0.414·20-s − 8.48·21-s − 2.82·22-s + 3.17·23-s − 2.82·24-s − 4.82·25-s − 6.82·26-s + 5.65·27-s − 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.63·3-s + 0.5·4-s − 0.185·5-s − 1.15·6-s − 1.13·7-s − 0.353·8-s + 1.66·9-s + 0.130·10-s + 0.852·11-s + 0.816·12-s + 1.89·13-s + 0.801·14-s − 0.302·15-s + 0.250·16-s + 1.17·17-s − 1.17·18-s − 0.838·19-s − 0.0926·20-s − 1.85·21-s − 0.603·22-s + 0.661·23-s − 0.577·24-s − 0.965·25-s − 1.33·26-s + 1.08·27-s − 0.566·28-s + ⋯

Functional equation

Λ(s)=(334s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(334s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 334334    =    2⋅1672 \cdot 167
Sign: 11
Analytic conductor: 2.667002.66700
Root analytic conductor: 1.633091.63309
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 334, ( :1/2), 1)(2,\ 334,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5578034731.557803473
L(12)L(\frac12) ≈\approx 1.5578034731.557803473
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+T 1 + T
167 1−T 1 - T
good3 1−2.82T+3T2 1 - 2.82T + 3T^{2}
5 1+0.414T+5T2 1 + 0.414T + 5T^{2}
7 1+3T+7T2 1 + 3T + 7T^{2}
11 1−2.82T+11T2 1 - 2.82T + 11T^{2}
13 1−6.82T+13T2 1 - 6.82T + 13T^{2}
17 1−4.82T+17T2 1 - 4.82T + 17T^{2}
19 1+3.65T+19T2 1 + 3.65T + 19T^{2}
23 1−3.17T+23T2 1 - 3.17T + 23T^{2}
29 1−1.17T+29T2 1 - 1.17T + 29T^{2}
31 1+7.82T+31T2 1 + 7.82T + 31T^{2}
37 1+7.24T+37T2 1 + 7.24T + 37T^{2}
41 1−0.343T+41T2 1 - 0.343T + 41T^{2}
43 1−2T+43T2 1 - 2T + 43T^{2}
47 1+11.4T+47T2 1 + 11.4T + 47T^{2}
53 1+9.58T+53T2 1 + 9.58T + 53T^{2}
59 1−6.41T+59T2 1 - 6.41T + 59T^{2}
61 1−8.48T+61T2 1 - 8.48T + 61T^{2}
67 1−0.414T+67T2 1 - 0.414T + 67T^{2}
71 1+2.48T+71T2 1 + 2.48T + 71T^{2}
73 1+13.6T+73T2 1 + 13.6T + 73T^{2}
79 1+8.48T+79T2 1 + 8.48T + 79T^{2}
83 1−14.8T+83T2 1 - 14.8T + 83T^{2}
89 1−8.65T+89T2 1 - 8.65T + 89T^{2}
97 1−3.82T+97T2 1 - 3.82T + 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

−11.40862124668015476417861597307, −10.27675099742077063652552279128, −9.427982635605966008212835267409, −8.804495338373476011558112604988, −8.105460500173576049875070469091, −7.01382435850729352939952586464, −6.07967418279830285609275530825, −3.72695237079035098531648530366, −3.30740996308260878574767635524, −1.62732020919446501342281036349, 1.62732020919446501342281036349, 3.30740996308260878574767635524, 3.72695237079035098531648530366, 6.07967418279830285609275530825, 7.01382435850729352939952586464, 8.105460500173576049875070469091, 8.804495338373476011558112604988, 9.427982635605966008212835267409, 10.27675099742077063652552279128, 11.40862124668015476417861597307

Graph of the ZZ-function along the critical line