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Properties

Label 2-960-15.14-c2-0-49
Degree 22
Conductor 960960
Sign 11
Analytic cond. 26.158126.1581
Root an. cond. 5.114495.11449
Motivic weight 22
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 − 5·5-s + 9·9-s − 15·15-s − 14·17-s + 22·19-s + 34·23-s + 25·25-s + 27·27-s + 2·31-s − 45·45-s − 14·47-s + 49·49-s − 42·51-s + 86·53-s + 66·57-s + 118·61-s + 102·69-s + 75·75-s + 98·79-s + 81·81-s − 154·83-s + 70·85-s + 6·93-s − 110·95-s − 106·107-s + 22·109-s + ⋯
L(s)  = 1  + 3-s − 5-s + 9-s − 15-s − 0.823·17-s + 1.15·19-s + 1.47·23-s + 25-s + 27-s + 2/31·31-s − 45-s − 0.297·47-s + 49-s − 0.823·51-s + 1.62·53-s + 1.15·57-s + 1.93·61-s + 1.47·69-s + 75-s + 1.24·79-s + 81-s − 1.85·83-s + 0.823·85-s + 2/31·93-s − 1.15·95-s − 0.990·107-s + 0.201·109-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+1) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 26.158126.1581
Root analytic conductor: 5.114495.11449
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: χ960(449,⋅)\chi_{960} (449, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 960, ( :1), 1)(2,\ 960,\ (\ :1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.4068622852.406862285
L(12)L(\frac12) ≈\approx 2.4068622852.406862285
L(2)L(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
3 1−pT 1 - p T
5 1+pT 1 + p T
good7 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
11 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
13 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
17 1+14T+p2T2 1 + 14 T + p^{2} T^{2}
19 1−22T+p2T2 1 - 22 T + p^{2} T^{2}
23 1−34T+p2T2 1 - 34 T + p^{2} T^{2}
29 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
31 1−2T+p2T2 1 - 2 T + p^{2} T^{2}
37 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
41 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
43 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
47 1+14T+p2T2 1 + 14 T + p^{2} T^{2}
53 1−86T+p2T2 1 - 86 T + p^{2} T^{2}
59 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
61 1−118T+p2T2 1 - 118 T + p^{2} T^{2}
67 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
71 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
73 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
79 1−98T+p2T2 1 - 98 T + p^{2} T^{2}
83 1+154T+p2T2 1 + 154 T + p^{2} T^{2}
89 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
97 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
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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

−9.650656002337281562464649995460, −8.856335304749391979132766625134, −8.268705780996585607452380261920, −7.28880001385535669069123991695, −6.88117166867882854200098686257, −5.26939005940498076423117612453, −4.30022219795605510970309119756, −3.44770007742856349128488020872, −2.53297145717665903502010451478, −0.964153572142294797891098416898, 0.964153572142294797891098416898, 2.53297145717665903502010451478, 3.44770007742856349128488020872, 4.30022219795605510970309119756, 5.26939005940498076423117612453, 6.88117166867882854200098686257, 7.28880001385535669069123991695, 8.268705780996585607452380261920, 8.856335304749391979132766625134, 9.650656002337281562464649995460

Graph of the ZZ-function along the critical line