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Properties

Label 2-7600-1.1-c1-0-131
Degree 22
Conductor 76007600
Sign 11
Analytic cond. 60.686360.6863
Root an. cond. 7.790147.79014
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  + 3.29·3-s + 1.78·7-s + 7.87·9-s + 5.08·11-s + 1.29·13-s − 0.213·17-s + 19-s + 5.89·21-s − 3.72·23-s + 16.0·27-s + 0.870·29-s + 16.7·33-s + 2·37-s + 4.27·39-s + 8.59·41-s − 3.67·43-s − 4.65·47-s − 3.80·49-s − 0.702·51-s − 11.0·53-s + 3.29·57-s + 4.70·59-s + 3.51·61-s + 14.0·63-s + 1.12·67-s − 12.2·69-s − 8.76·71-s + ⋯
L(s)  = 1  + 1.90·3-s + 0.675·7-s + 2.62·9-s + 1.53·11-s + 0.359·13-s − 0.0517·17-s + 0.229·19-s + 1.28·21-s − 0.776·23-s + 3.09·27-s + 0.161·29-s + 2.91·33-s + 0.328·37-s + 0.684·39-s + 1.34·41-s − 0.560·43-s − 0.679·47-s − 0.543·49-s − 0.0984·51-s − 1.51·53-s + 0.436·57-s + 0.612·59-s + 0.449·61-s + 1.77·63-s + 0.137·67-s − 1.47·69-s − 1.03·71-s + ⋯

Functional equation

Λ(s)=(7600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76007600    =    24⋅52⋅192^{4} \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 60.686360.6863
Root analytic conductor: 7.790147.79014
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7600, ( :1/2), 1)(2,\ 7600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.7248389635.724838963
L(12)L(\frac12) ≈\approx 5.7248389635.724838963
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
5 1 1
19 1−T 1 - T
good3 1−3.29T+3T2 1 - 3.29T + 3T^{2}
7 1−1.78T+7T2 1 - 1.78T + 7T^{2}
11 1−5.08T+11T2 1 - 5.08T + 11T^{2}
13 1−1.29T+13T2 1 - 1.29T + 13T^{2}
17 1+0.213T+17T2 1 + 0.213T + 17T^{2}
23 1+3.72T+23T2 1 + 3.72T + 23T^{2}
29 1−0.870T+29T2 1 - 0.870T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1−8.59T+41T2 1 - 8.59T + 41T^{2}
43 1+3.67T+43T2 1 + 3.67T + 43T^{2}
47 1+4.65T+47T2 1 + 4.65T + 47T^{2}
53 1+11.0T+53T2 1 + 11.0T + 53T^{2}
59 1−4.70T+59T2 1 - 4.70T + 59T^{2}
61 1−3.51T+61T2 1 - 3.51T + 61T^{2}
67 1−1.12T+67T2 1 - 1.12T + 67T^{2}
71 1+8.76T+71T2 1 + 8.76T + 71T^{2}
73 1+6.80T+73T2 1 + 6.80T + 73T^{2}
79 1+14.5T+79T2 1 + 14.5T + 79T^{2}
83 1+9.74T+83T2 1 + 9.74T + 83T^{2}
89 1+6.76T+89T2 1 + 6.76T + 89T^{2}
97 1+4.16T+97T2 1 + 4.16T + 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

−8.025351612967020190845123725778, −7.38793990890304135597963632068, −6.71585449435198832799473612615, −5.93397952149294345636170582796, −4.64323506428044266116680475868, −4.16972584032069745082670453469, −3.50075036861646334228878811282, −2.75298921261753567873681177743, −1.76832288348874321548023722240, −1.28788048919146738382091250573, 1.28788048919146738382091250573, 1.76832288348874321548023722240, 2.75298921261753567873681177743, 3.50075036861646334228878811282, 4.16972584032069745082670453469, 4.64323506428044266116680475868, 5.93397952149294345636170582796, 6.71585449435198832799473612615, 7.38793990890304135597963632068, 8.025351612967020190845123725778

Graph of the ZZ-function along the critical line