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Properties

Label 2-960-8.5-c1-0-0
Degree 22
Conductor 960960
Sign −0.965−0.258i-0.965 - 0.258i
Analytic cond. 7.665637.66563
Root an. cond. 2.768682.76868
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − i·5-s − 2·7-s − 9-s + 1.46i·11-s − 1.46i·13-s + 15-s − 3.46·17-s + 6.92i·19-s − 2i·21-s − 4·23-s − 25-s − i·27-s + 4.92i·29-s − 7.46·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.447i·5-s − 0.755·7-s − 0.333·9-s + 0.441i·11-s − 0.406i·13-s + 0.258·15-s − 0.840·17-s + 1.58i·19-s − 0.436i·21-s − 0.834·23-s − 0.200·25-s − 0.192i·27-s + 0.915i·29-s − 1.34·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((−0.965−0.258i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+1/2) L(s)=((−0.965−0.258i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: −0.965−0.258i-0.965 - 0.258i
Analytic conductor: 7.665637.66563
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ960(481,⋅)\chi_{960} (481, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :1/2), −0.965−0.258i)(2,\ 960,\ (\ :1/2),\ -0.965 - 0.258i)

Particular Values

L(1)L(1) ≈\approx 0.0561921+0.426821i0.0561921 + 0.426821i
L(12)L(\frac12) ≈\approx 0.0561921+0.426821i0.0561921 + 0.426821i
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−iT 1 - iT
5 1+iT 1 + iT
good7 1+2T+7T2 1 + 2T + 7T^{2}
11 1−1.46iT−11T2 1 - 1.46iT - 11T^{2}
13 1+1.46iT−13T2 1 + 1.46iT - 13T^{2}
17 1+3.46T+17T2 1 + 3.46T + 17T^{2}
19 1−6.92iT−19T2 1 - 6.92iT - 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−4.92iT−29T2 1 - 4.92iT - 29T^{2}
31 1+7.46T+31T2 1 + 7.46T + 31T^{2}
37 1+2.53iT−37T2 1 + 2.53iT - 37T^{2}
41 1+8.92T+41T2 1 + 8.92T + 41T^{2}
43 1−6.92iT−43T2 1 - 6.92iT - 43T^{2}
47 1−4T+47T2 1 - 4T + 47T^{2}
53 1+12.9iT−53T2 1 + 12.9iT - 53T^{2}
59 1−2.53iT−59T2 1 - 2.53iT - 59T^{2}
61 1−4iT−61T2 1 - 4iT - 61T^{2}
67 1−6.92iT−67T2 1 - 6.92iT - 67T^{2}
71 1+6.92T+71T2 1 + 6.92T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1−6.39T+79T2 1 - 6.39T + 79T^{2}
83 1+8iT−83T2 1 + 8iT - 83T^{2}
89 1+12.9T+89T2 1 + 12.9T + 89T^{2}
97 1−0.928T+97T2 1 - 0.928T + 97T^{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

−10.19485394336769475921239115015, −9.708826610739511848773486215761, −8.823961227443358454416612792019, −8.057002644153694982699164863221, −6.99121605412110690812953763470, −6.00275596419484365203390871301, −5.20202905090805955754151396484, −4.11004329381752899349374828778, −3.33007010539779106782583254286, −1.86095555405810258373922402236, 0.18572842624847487243850789688, 2.06663045020158302442231631586, 3.06127327217362228335623237215, 4.16650753041499254463902367224, 5.45019366981643563032793904104, 6.49068351820525679154532060251, 6.88227941416571216392276642571, 7.88792197553860943065143877808, 8.898957335225494009222580641807, 9.503994028980192511300549039624

Graph of the ZZ-function along the critical line