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Properties

Label 2-960-1.1-c1-0-6
Degree 22
Conductor 960960
Sign 11
Analytic cond. 7.665637.66563
Root an. cond. 2.768682.76868
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-s + 5-s + 4·7-s + 9-s + 4·11-s − 6·13-s − 15-s + 2·17-s + 4·19-s − 4·21-s + 25-s − 27-s − 10·29-s + 4·31-s − 4·33-s + 4·35-s + 10·37-s + 6·39-s + 2·41-s − 4·43-s + 45-s − 8·47-s + 9·49-s − 2·51-s − 2·53-s + 4·55-s − 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s − 0.696·33-s + 0.676·35-s + 1.64·37-s + 0.960·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.274·53-s + 0.539·55-s − 0.529·57-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 7.665637.66563
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 960, ( :1/2), 1)(2,\ 960,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7431258031.743125803
L(12)L(\frac12) ≈\approx 1.7431258031.743125803
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
3 1+T 1 + T
5 1−T 1 - T
good7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+10T+pT2 1 + 10 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−9.758345846761255680723166486652, −9.569887168096234669182184873626, −8.218385815535771240939930654544, −7.49235926303250200960974861538, −6.65509689812873112624861887690, −5.45857154341122253575185626745, −4.97527404703815748732112348741, −3.93952361309394612597846303015, −2.28452944523597307623702961432, −1.18879387693747021147526651062, 1.18879387693747021147526651062, 2.28452944523597307623702961432, 3.93952361309394612597846303015, 4.97527404703815748732112348741, 5.45857154341122253575185626745, 6.65509689812873112624861887690, 7.49235926303250200960974861538, 8.218385815535771240939930654544, 9.569887168096234669182184873626, 9.758345846761255680723166486652

Graph of the ZZ-function along the critical line