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Properties

Label 2-960-5.3-c2-0-20
Degree 22
Conductor 960960
Sign 0.130−0.991i0.130 - 0.991i
Analytic cond. 26.158126.1581
Root an. cond. 5.114495.11449
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)3-s + (4.67 + 1.77i)5-s + (3.44 − 3.44i)7-s + 2.99i·9-s − 11.3·11-s + (5.55 + 5.55i)13-s + (3.55 + 7.89i)15-s + (−17.3 + 17.3i)17-s + 8.69i·19-s + 8.44·21-s + (11.5 + 11.5i)23-s + (18.6 + 16.5i)25-s + (−3.67 + 3.67i)27-s + 35.1i·29-s + 10.6·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.934 + 0.355i)5-s + (0.492 − 0.492i)7-s + 0.333i·9-s − 1.03·11-s + (0.426 + 0.426i)13-s + (0.236 + 0.526i)15-s + (−1.02 + 1.02i)17-s + 0.457i·19-s + 0.402·21-s + (0.502 + 0.502i)23-s + (0.747 + 0.663i)25-s + (−0.136 + 0.136i)27-s + 1.21i·29-s + 0.345·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.130−0.991i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+1) L(s)=((0.130−0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.130−0.991i0.130 - 0.991i
Analytic conductor: 26.158126.1581
Root analytic conductor: 5.114495.11449
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ960(193,⋅)\chi_{960} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :1), 0.130−0.991i)(2,\ 960,\ (\ :1),\ 0.130 - 0.991i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.4490237252.449023725
L(12)L(\frac12) ≈\approx 2.4490237252.449023725
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+(−1.22−1.22i)T 1 + (-1.22 - 1.22i)T
5 1+(−4.67−1.77i)T 1 + (-4.67 - 1.77i)T
good7 1+(−3.44+3.44i)T−49iT2 1 + (-3.44 + 3.44i)T - 49iT^{2}
11 1+11.3T+121T2 1 + 11.3T + 121T^{2}
13 1+(−5.55−5.55i)T+169iT2 1 + (-5.55 - 5.55i)T + 169iT^{2}
17 1+(17.3−17.3i)T−289iT2 1 + (17.3 - 17.3i)T - 289iT^{2}
19 1−8.69iT−361T2 1 - 8.69iT - 361T^{2}
23 1+(−11.5−11.5i)T+529iT2 1 + (-11.5 - 11.5i)T + 529iT^{2}
29 1−35.1iT−841T2 1 - 35.1iT - 841T^{2}
31 1−10.6T+961T2 1 - 10.6T + 961T^{2}
37 1+(−6.04+6.04i)T−1.36e3iT2 1 + (-6.04 + 6.04i)T - 1.36e3iT^{2}
41 1−0.696T+1.68e3T2 1 - 0.696T + 1.68e3T^{2}
43 1+(−26.4−26.4i)T+1.84e3iT2 1 + (-26.4 - 26.4i)T + 1.84e3iT^{2}
47 1+(−44.2+44.2i)T−2.20e3iT2 1 + (-44.2 + 44.2i)T - 2.20e3iT^{2}
53 1+(−0.696−0.696i)T+2.80e3iT2 1 + (-0.696 - 0.696i)T + 2.80e3iT^{2}
59 1+39.9iT−3.48e3T2 1 + 39.9iT - 3.48e3T^{2}
61 1+5.90T+3.72e3T2 1 + 5.90T + 3.72e3T^{2}
67 1+(−45.1+45.1i)T−4.48e3iT2 1 + (-45.1 + 45.1i)T - 4.48e3iT^{2}
71 1+68T+5.04e3T2 1 + 68T + 5.04e3T^{2}
73 1+(−77.7−77.7i)T+5.32e3iT2 1 + (-77.7 - 77.7i)T + 5.32e3iT^{2}
79 1−24.4iT−6.24e3T2 1 - 24.4iT - 6.24e3T^{2}
83 1+(13.1+13.1i)T+6.88e3iT2 1 + (13.1 + 13.1i)T + 6.88e3iT^{2}
89 1+82.1iT−7.92e3T2 1 + 82.1iT - 7.92e3T^{2}
97 1+(24.5−24.5i)T−9.40e3iT2 1 + (24.5 - 24.5i)T - 9.40e3iT^{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

−10.12970969084998704124095958282, −9.184317258188158320405609506988, −8.476610197094911068634084675891, −7.55094784516361353761511570210, −6.61546980019509243437712428500, −5.64448852254817428679515064569, −4.75127434454207242969339712386, −3.70413043769854087468342690817, −2.53687835867811447663255922298, −1.52344328202321822850418770658, 0.73423915449271714187503884576, 2.23091509852329864483018942935, 2.74916042269715485301059886576, 4.49021250771653457460045267612, 5.31547067562833720662951148501, 6.14397470348236770145803749560, 7.14162957565806170358947381981, 8.108942835212444258501246460112, 8.802022057006683479429526313448, 9.462693901985571961969736609145

Graph of the ZZ-function along the critical line