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Properties

Label 2-960-20.7-c1-0-17
Degree 22
Conductor 960960
Sign −0.197+0.980i-0.197 + 0.980i
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  + (−0.707 − 0.707i)3-s + (−2.23 + 0.0743i)5-s + (3.16 − 3.16i)7-s + 1.00i·9-s + 4.46i·11-s + (2.51 − 2.51i)13-s + (1.63 + 1.52i)15-s + (−2.30 − 2.30i)17-s + 2.61·19-s − 4.46·21-s + (−1.64 − 1.64i)23-s + (4.98 − 0.332i)25-s + (0.707 − 0.707i)27-s − 8.17i·29-s + 4i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.999 + 0.0332i)5-s + (1.19 − 1.19i)7-s + 0.333i·9-s + 1.34i·11-s + (0.698 − 0.698i)13-s + (0.421 + 0.394i)15-s + (−0.560 − 0.560i)17-s + 0.600·19-s − 0.975·21-s + (−0.342 − 0.342i)23-s + (0.997 − 0.0664i)25-s + (0.136 − 0.136i)27-s − 1.51i·29-s + 0.718i·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((−0.197+0.980i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+1/2) L(s)=((−0.197+0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: −0.197+0.980i-0.197 + 0.980i
Analytic conductor: 7.665637.66563
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ960(127,⋅)\chi_{960} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :1/2), −0.197+0.980i)(2,\ 960,\ (\ :1/2),\ -0.197 + 0.980i)

Particular Values

L(1)L(1) ≈\approx 0.703746−0.859477i0.703746 - 0.859477i
L(12)L(\frac12) ≈\approx 0.703746−0.859477i0.703746 - 0.859477i
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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
5 1+(2.23−0.0743i)T 1 + (2.23 - 0.0743i)T
good7 1+(−3.16+3.16i)T−7iT2 1 + (-3.16 + 3.16i)T - 7iT^{2}
11 1−4.46iT−11T2 1 - 4.46iT - 11T^{2}
13 1+(−2.51+2.51i)T−13iT2 1 + (-2.51 + 2.51i)T - 13iT^{2}
17 1+(2.30+2.30i)T+17iT2 1 + (2.30 + 2.30i)T + 17iT^{2}
19 1−2.61T+19T2 1 - 2.61T + 19T^{2}
23 1+(1.64+1.64i)T+23iT2 1 + (1.64 + 1.64i)T + 23iT^{2}
29 1+8.17iT−29T2 1 + 8.17iT - 29T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
37 1+(5.80+5.80i)T+37iT2 1 + (5.80 + 5.80i)T + 37iT^{2}
41 1+2.61T+41T2 1 + 2.61T + 41T^{2}
43 1+(5.14+5.14i)T+43iT2 1 + (5.14 + 5.14i)T + 43iT^{2}
47 1+(0.679−0.679i)T−47iT2 1 + (0.679 - 0.679i)T - 47iT^{2}
53 1+(−7.81+7.81i)T−53iT2 1 + (-7.81 + 7.81i)T - 53iT^{2}
59 1−4.88T+59T2 1 - 4.88T + 59T^{2}
61 1+12.2T+61T2 1 + 12.2T + 61T^{2}
67 1+(−9.44+9.44i)T−67iT2 1 + (-9.44 + 9.44i)T - 67iT^{2}
71 1+5.65iT−71T2 1 + 5.65iT - 71T^{2}
73 1+(5.61−5.61i)T−73iT2 1 + (5.61 - 5.61i)T - 73iT^{2}
79 1−3.57T+79T2 1 - 3.57T + 79T^{2}
83 1+(−1.34−1.34i)T+83iT2 1 + (-1.34 - 1.34i)T + 83iT^{2}
89 1+17.6iT−89T2 1 + 17.6iT - 89T^{2}
97 1+(1.32+1.32i)T+97iT2 1 + (1.32 + 1.32i)T + 97iT^{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

−10.11554480867543963573246022045, −8.706497817599969886070746937079, −7.85125885345038956262024146903, −7.38631043797840178535730385539, −6.67910141701796140906758451523, −5.18482845237374683992105777920, −4.52026210868024488349177390245, −3.64879607809458234361577461822, −1.93546302058491607081430974221, −0.59854138231452478268799427676, 1.44073051550183293578680039389, 3.09460265800754781096187210456, 4.07223776046739447925817503558, 5.06014328972212870665761501087, 5.78614924605767960986165757100, 6.79202809380029193145178164342, 8.063731809879568541342581687618, 8.575549994594952268886533131640, 9.126776187764919180585604854745, 10.57821360115196087652278996867

Graph of the ZZ-function along the critical line