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Properties

Label 2-7616-1.1-c1-0-105
Degree 22
Conductor 76167616
Sign 11
Analytic cond. 60.814060.8140
Root an. cond. 7.798337.79833
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  + 1.90·3-s + 2.90·5-s − 7-s + 0.618·9-s + 2.79·11-s + 1.95·13-s + 5.52·15-s + 17-s − 4.42·19-s − 1.90·21-s + 1.72·23-s + 3.42·25-s − 4.53·27-s − 8.15·29-s + 9.64·31-s + 5.31·33-s − 2.90·35-s + 2.61·37-s + 3.71·39-s − 1.43·41-s + 8.99·43-s + 1.79·45-s + 2.65·47-s + 49-s + 1.90·51-s + 4.50·53-s + 8.10·55-s + ⋯
L(s)  = 1  + 1.09·3-s + 1.29·5-s − 0.377·7-s + 0.206·9-s + 0.842·11-s + 0.541·13-s + 1.42·15-s + 0.242·17-s − 1.01·19-s − 0.415·21-s + 0.360·23-s + 0.684·25-s − 0.871·27-s − 1.51·29-s + 1.73·31-s + 0.925·33-s − 0.490·35-s + 0.429·37-s + 0.594·39-s − 0.224·41-s + 1.37·43-s + 0.267·45-s + 0.387·47-s + 0.142·49-s + 0.266·51-s + 0.618·53-s + 1.09·55-s + ⋯

Functional equation

Λ(s)=(7616s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7616s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76167616    =    26⋅7⋅172^{6} \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 60.814060.8140
Root analytic conductor: 7.798337.79833
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7616, ( :1/2), 1)(2,\ 7616,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.1635831634.163583163
L(12)L(\frac12) ≈\approx 4.1635831634.163583163
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 1
7 1+T 1 + T
17 1−T 1 - T
good3 1−1.90T+3T2 1 - 1.90T + 3T^{2}
5 1−2.90T+5T2 1 - 2.90T + 5T^{2}
11 1−2.79T+11T2 1 - 2.79T + 11T^{2}
13 1−1.95T+13T2 1 - 1.95T + 13T^{2}
19 1+4.42T+19T2 1 + 4.42T + 19T^{2}
23 1−1.72T+23T2 1 - 1.72T + 23T^{2}
29 1+8.15T+29T2 1 + 8.15T + 29T^{2}
31 1−9.64T+31T2 1 - 9.64T + 31T^{2}
37 1−2.61T+37T2 1 - 2.61T + 37T^{2}
41 1+1.43T+41T2 1 + 1.43T + 41T^{2}
43 1−8.99T+43T2 1 - 8.99T + 43T^{2}
47 1−2.65T+47T2 1 - 2.65T + 47T^{2}
53 1−4.50T+53T2 1 - 4.50T + 53T^{2}
59 1+12.2T+59T2 1 + 12.2T + 59T^{2}
61 1−13.8T+61T2 1 - 13.8T + 61T^{2}
67 1−11.8T+67T2 1 - 11.8T + 67T^{2}
71 1−8.64T+71T2 1 - 8.64T + 71T^{2}
73 1−2.37T+73T2 1 - 2.37T + 73T^{2}
79 1−8.59T+79T2 1 - 8.59T + 79T^{2}
83 1−10.8T+83T2 1 - 10.8T + 83T^{2}
89 1+14.0T+89T2 1 + 14.0T + 89T^{2}
97 1−14.0T+97T2 1 - 14.0T + 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

−8.052339268933643340096031021568, −7.18617137714968631093070196980, −6.30742426914204626502146480120, −6.04408708699242834405831665720, −5.13195312169683749382178793611, −4.05129539922926690655285107763, −3.50815935918370658877403873936, −2.49840489940042788632842785416, −2.07153974051976643135584399764, −1.00029120283349766712854490917, 1.00029120283349766712854490917, 2.07153974051976643135584399764, 2.49840489940042788632842785416, 3.50815935918370658877403873936, 4.05129539922926690655285107763, 5.13195312169683749382178793611, 6.04408708699242834405831665720, 6.30742426914204626502146480120, 7.18617137714968631093070196980, 8.052339268933643340096031021568

Graph of the ZZ-function along the critical line