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Properties

Label 2-740-1.1-c1-0-6
Degree 22
Conductor 740740
Sign 11
Analytic cond. 5.908925.90892
Root an. cond. 2.430822.43082
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 − 3·7-s + 6·9-s + 5·11-s + 2·13-s − 3·15-s + 4·17-s − 4·19-s − 9·21-s + 6·23-s + 25-s + 9·27-s + 6·29-s − 4·31-s + 15·33-s + 3·35-s − 37-s + 6·39-s − 9·41-s + 10·43-s − 6·45-s − 11·47-s + 2·49-s + 12·51-s − 11·53-s − 5·55-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s − 1.13·7-s + 2·9-s + 1.50·11-s + 0.554·13-s − 0.774·15-s + 0.970·17-s − 0.917·19-s − 1.96·21-s + 1.25·23-s + 1/5·25-s + 1.73·27-s + 1.11·29-s − 0.718·31-s + 2.61·33-s + 0.507·35-s − 0.164·37-s + 0.960·39-s − 1.40·41-s + 1.52·43-s − 0.894·45-s − 1.60·47-s + 2/7·49-s + 1.68·51-s − 1.51·53-s − 0.674·55-s + ⋯

Functional equation

Λ(s)=(740s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(740s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 740740    =    22⋅5⋅372^{2} \cdot 5 \cdot 37
Sign: 11
Analytic conductor: 5.908925.90892
Root analytic conductor: 2.430822.43082
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 740, ( :1/2), 1)(2,\ 740,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5349965952.534996595
L(12)L(\frac12) ≈\approx 2.5349965952.534996595
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
37 1+T 1 + T
good3 1−pT+pT2 1 - p T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+11T+pT2 1 + 11 T + p T^{2}
53 1+11T+pT2 1 + 11 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+9T+pT2 1 + 9 T + p T^{2}
89 1+16T+pT2 1 + 16 T + p T^{2}
97 1−12T+pT2 1 - 12 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

−10.03174952203399649378141239133, −9.284993923943248760027735739396, −8.798405749374759083599131072327, −7.965299355909049566818828913125, −6.96228910991813633687344598913, −6.31311251034846705906848780783, −4.48878031439686602633803909869, −3.49416389960600763793673657644, −3.08976764586911151057853710056, −1.47616131815811688268284937786, 1.47616131815811688268284937786, 3.08976764586911151057853710056, 3.49416389960600763793673657644, 4.48878031439686602633803909869, 6.31311251034846705906848780783, 6.96228910991813633687344598913, 7.965299355909049566818828913125, 8.798405749374759083599131072327, 9.284993923943248760027735739396, 10.03174952203399649378141239133

Graph of the ZZ-function along the critical line