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Properties

Label 2-7360-1.1-c1-0-134
Degree 22
Conductor 73607360
Sign −1-1
Analytic cond. 58.769858.7698
Root an. cond. 7.666157.66615
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·9-s − 2·11-s + 5·13-s − 15-s − 4·17-s + 2·19-s − 23-s + 25-s − 5·27-s + 3·29-s + 7·31-s − 2·33-s + 2·37-s + 5·39-s − 9·41-s + 4·43-s + 2·45-s − 9·47-s − 7·49-s − 4·51-s + 6·53-s + 2·55-s + 2·57-s − 2·61-s − 5·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.603·11-s + 1.38·13-s − 0.258·15-s − 0.970·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 1.25·31-s − 0.348·33-s + 0.328·37-s + 0.800·39-s − 1.40·41-s + 0.609·43-s + 0.298·45-s − 1.31·47-s − 49-s − 0.560·51-s + 0.824·53-s + 0.269·55-s + 0.264·57-s − 0.256·61-s − 0.620·65-s + ⋯

Functional equation

Λ(s)=(7360s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7360s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 73607360    =    26⋅5⋅232^{6} \cdot 5 \cdot 23
Sign: −1-1
Analytic conductor: 58.769858.7698
Root analytic conductor: 7.666157.66615
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7360, ( :1/2), −1)(2,\ 7360,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+T 1 + T
23 1+T 1 + T
good3 1−T+pT2 1 - T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1−7T+pT2 1 - 7 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+T+pT2 1 + T + p T^{2}
73 1−T+pT2 1 - T + p T^{2}
79 1+14T+pT2 1 + 14 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−16T+pT2 1 - 16 T + p T^{2}
97 1+4T+pT2 1 + 4 T + p T^{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

−7.87772245704560807235754252641, −6.78736577097464942111263138964, −6.28390217387550107725218900415, −5.43514666489651177446009426067, −4.64513196673738552103167532966, −3.81533425599749720944212788533, −3.12100944634508531847354355429, −2.44972737007574203809585963501, −1.30944185091218128198182897594, 0, 1.30944185091218128198182897594, 2.44972737007574203809585963501, 3.12100944634508531847354355429, 3.81533425599749720944212788533, 4.64513196673738552103167532966, 5.43514666489651177446009426067, 6.28390217387550107725218900415, 6.78736577097464942111263138964, 7.87772245704560807235754252641

Graph of the ZZ-function along the critical line