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Properties

Label 2-7360-1.1-c1-0-31
Degree 22
Conductor 73607360
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5-s + 2·7-s + 6·9-s − 13-s + 3·15-s − 6·21-s − 23-s + 25-s − 9·27-s + 3·29-s − 3·31-s − 2·35-s + 8·37-s + 3·39-s + 3·41-s − 2·43-s − 6·45-s + 11·47-s − 3·49-s + 14·53-s − 8·59-s + 4·61-s + 12·63-s + 65-s − 4·67-s + 3·69-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 0.755·7-s + 2·9-s − 0.277·13-s + 0.774·15-s − 1.30·21-s − 0.208·23-s + 1/5·25-s − 1.73·27-s + 0.557·29-s − 0.538·31-s − 0.338·35-s + 1.31·37-s + 0.480·39-s + 0.468·41-s − 0.304·43-s − 0.894·45-s + 1.60·47-s − 3/7·49-s + 1.92·53-s − 1.04·59-s + 0.512·61-s + 1.51·63-s + 0.124·65-s − 0.488·67-s + 0.361·69-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: 11
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: 00
Selberg data: (2, 7360, ( :1/2), 1)(2,\ 7360,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.91246876470.9124687647
L(12)L(\frac12) ≈\approx 0.91246876470.9124687647
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+pT+pT2 1 + p T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1−11T+pT2 1 - 11 T + p T^{2}
53 1−14T+pT2 1 - 14 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1−4T+pT2 1 - 4 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+7T+pT2 1 + 7 T + p T^{2}
73 1+9T+pT2 1 + 9 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1−18T+pT2 1 - 18 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

−7.61097695431891812572771540551, −7.23894765301243027339221170097, −6.36358694667981234706594365623, −5.78404075969593536096693616107, −5.12820814876093260820687152704, −4.49132162780431694098830286455, −3.94355360286602801503349744631, −2.60142185594210330236613786748, −1.44280074660581855359414549843, −0.56923778323449802951397178654, 0.56923778323449802951397178654, 1.44280074660581855359414549843, 2.60142185594210330236613786748, 3.94355360286602801503349744631, 4.49132162780431694098830286455, 5.12820814876093260820687152704, 5.78404075969593536096693616107, 6.36358694667981234706594365623, 7.23894765301243027339221170097, 7.61097695431891812572771540551

Graph of the ZZ-function along the critical line