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Properties

Label 2-960-20.7-c1-0-4
Degree 22
Conductor 960960
Sign −0.899−0.437i-0.899 - 0.437i
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 + (−0.489 + 2.18i)5-s + (−0.692 + 0.692i)7-s + 1.00i·9-s + 0.979i·11-s + (−3.49 + 3.49i)13-s + (−1.88 + 1.19i)15-s + (−2.67 − 2.67i)17-s + 3.34·19-s − 0.979·21-s + (−3.80 − 3.80i)23-s + (−4.52 − 2.13i)25-s + (−0.707 + 0.707i)27-s + 3.74i·29-s + 4i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.218 + 0.975i)5-s + (−0.261 + 0.261i)7-s + 0.333i·9-s + 0.295i·11-s + (−0.970 + 0.970i)13-s + (−0.487 + 0.308i)15-s + (−0.647 − 0.647i)17-s + 0.766·19-s − 0.213·21-s + (−0.793 − 0.793i)23-s + (−0.904 − 0.427i)25-s + (−0.136 + 0.136i)27-s + 0.696i·29-s + 0.718i·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((−0.899−0.437i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+1/2) L(s)=((−0.899−0.437i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: −0.899−0.437i-0.899 - 0.437i
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), −0.899−0.437i)(2,\ 960,\ (\ :1/2),\ -0.899 - 0.437i)

Particular Values

L(1)L(1) ≈\approx 0.241133+1.04745i0.241133 + 1.04745i
L(12)L(\frac12) ≈\approx 0.241133+1.04745i0.241133 + 1.04745i
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+(0.489−2.18i)T 1 + (0.489 - 2.18i)T
good7 1+(0.692−0.692i)T−7iT2 1 + (0.692 - 0.692i)T - 7iT^{2}
11 1−0.979iT−11T2 1 - 0.979iT - 11T^{2}
13 1+(3.49−3.49i)T−13iT2 1 + (3.49 - 3.49i)T - 13iT^{2}
17 1+(2.67+2.67i)T+17iT2 1 + (2.67 + 2.67i)T + 17iT^{2}
19 1−3.34T+19T2 1 - 3.34T + 19T^{2}
23 1+(3.80+3.80i)T+23iT2 1 + (3.80 + 3.80i)T + 23iT^{2}
29 1−3.74iT−29T2 1 - 3.74iT - 29T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
37 1+(4.11+4.11i)T+37iT2 1 + (4.11 + 4.11i)T + 37iT^{2}
41 1+3.34T+41T2 1 + 3.34T + 41T^{2}
43 1+(−8.21−8.21i)T+43iT2 1 + (-8.21 - 8.21i)T + 43iT^{2}
47 1+(−9.19+9.19i)T−47iT2 1 + (-9.19 + 9.19i)T - 47iT^{2}
53 1+(7.34−7.34i)T−53iT2 1 + (7.34 - 7.34i)T - 53iT^{2}
59 1+11.3T+59T2 1 + 11.3T + 59T^{2}
61 1+1.68T+61T2 1 + 1.68T + 61T^{2}
67 1+(−4.51+4.51i)T−67iT2 1 + (-4.51 + 4.51i)T - 67iT^{2}
71 1−5.65iT−71T2 1 - 5.65iT - 71T^{2}
73 1+(6.34−6.34i)T−73iT2 1 + (6.34 - 6.34i)T - 73iT^{2}
79 1−16.3T+79T2 1 - 16.3T + 79T^{2}
83 1+(4.91+4.91i)T+83iT2 1 + (4.91 + 4.91i)T + 83iT^{2}
89 1−12.6iT−89T2 1 - 12.6iT - 89T^{2}
97 1+(−6.38−6.38i)T+97iT2 1 + (-6.38 - 6.38i)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

−10.35602714572487020209217958250, −9.487618455469444510078292169020, −8.970133904857321543436862355530, −7.70811464512400583341889790871, −7.09730545231708826423991195701, −6.26804303203317932239052736942, −4.98915553513101434659995378836, −4.10869803107703551251500265463, −2.98456462461537860111860472508, −2.18147062189300053246221555894, 0.44874906152874954872764080781, 1.91859151830482869388918488735, 3.25590854521229980881805712197, 4.25598821670320177750176298891, 5.33271967963995559407952306434, 6.16868069353769778178165174728, 7.47574482251521427865830322329, 7.87916994497391140027590366070, 8.806141776581272841213841139244, 9.577440928128243345534759624903

Graph of the ZZ-function along the critical line