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Properties

Label 2-608-1.1-c1-0-13
Degree 22
Conductor 608608
Sign 11
Analytic cond. 4.854904.85490
Root an. cond. 2.203382.20338
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  + 2.56·3-s + 1.56·5-s + 3·7-s + 3.56·9-s − 3.56·11-s + 2.56·13-s + 4·15-s − 8.12·17-s + 19-s + 7.68·21-s + 1.43·23-s − 2.56·25-s + 1.43·27-s − 7.68·29-s + 0.876·31-s − 9.12·33-s + 4.68·35-s − 1.12·37-s + 6.56·39-s + 4·41-s + 9.56·43-s + 5.56·45-s + 8.68·47-s + 2·49-s − 20.8·51-s − 8.56·53-s − 5.56·55-s + ⋯
L(s)  = 1  + 1.47·3-s + 0.698·5-s + 1.13·7-s + 1.18·9-s − 1.07·11-s + 0.710·13-s + 1.03·15-s − 1.97·17-s + 0.229·19-s + 1.67·21-s + 0.299·23-s − 0.512·25-s + 0.276·27-s − 1.42·29-s + 0.157·31-s − 1.58·33-s + 0.791·35-s − 0.184·37-s + 1.05·39-s + 0.624·41-s + 1.45·43-s + 0.829·45-s + 1.26·47-s + 0.285·49-s − 2.91·51-s − 1.17·53-s − 0.749·55-s + ⋯

Functional equation

Λ(s)=(608s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(608s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 608608    =    25⋅192^{5} \cdot 19
Sign: 11
Analytic conductor: 4.854904.85490
Root analytic conductor: 2.203382.20338
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 608, ( :1/2), 1)(2,\ 608,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7220462122.722046212
L(12)L(\frac12) ≈\approx 2.7220462122.722046212
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
19 1−T 1 - T
good3 1−2.56T+3T2 1 - 2.56T + 3T^{2}
5 1−1.56T+5T2 1 - 1.56T + 5T^{2}
7 1−3T+7T2 1 - 3T + 7T^{2}
11 1+3.56T+11T2 1 + 3.56T + 11T^{2}
13 1−2.56T+13T2 1 - 2.56T + 13T^{2}
17 1+8.12T+17T2 1 + 8.12T + 17T^{2}
23 1−1.43T+23T2 1 - 1.43T + 23T^{2}
29 1+7.68T+29T2 1 + 7.68T + 29T^{2}
31 1−0.876T+31T2 1 - 0.876T + 31T^{2}
37 1+1.12T+37T2 1 + 1.12T + 37T^{2}
41 1−4T+41T2 1 - 4T + 41T^{2}
43 1−9.56T+43T2 1 - 9.56T + 43T^{2}
47 1−8.68T+47T2 1 - 8.68T + 47T^{2}
53 1+8.56T+53T2 1 + 8.56T + 53T^{2}
59 1+8.56T+59T2 1 + 8.56T + 59T^{2}
61 1+5.80T+61T2 1 + 5.80T + 61T^{2}
67 1+4.56T+67T2 1 + 4.56T + 67T^{2}
71 1−12.2T+71T2 1 - 12.2T + 71T^{2}
73 1−7.24T+73T2 1 - 7.24T + 73T^{2}
79 1−10T+79T2 1 - 10T + 79T^{2}
83 1−7.36T+83T2 1 - 7.36T + 83T^{2}
89 1−9.36T+89T2 1 - 9.36T + 89T^{2}
97 1+1.12T+97T2 1 + 1.12T + 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

−10.76049969597573691655238784929, −9.392410201670798868356218425549, −8.980582610518813772532807858459, −8.034577044975961632511233168401, −7.49629067270974422852508811169, −6.11417336287529391942224171467, −4.96591759499945842784207650948, −3.91433566893515775066259090814, −2.53346238926307565915139486861, −1.85961794065305034181282345647, 1.85961794065305034181282345647, 2.53346238926307565915139486861, 3.91433566893515775066259090814, 4.96591759499945842784207650948, 6.11417336287529391942224171467, 7.49629067270974422852508811169, 8.034577044975961632511233168401, 8.980582610518813772532807858459, 9.392410201670798868356218425549, 10.76049969597573691655238784929

Graph of the ZZ-function along the critical line