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Properties

Label 2-3960-5.4-c1-0-47
Degree 22
Conductor 39603960
Sign 0.825+0.564i0.825 + 0.564i
Analytic cond. 31.620731.6207
Root an. cond. 5.623235.62323
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  + (−1.26 + 1.84i)5-s + 3.12i·7-s + 11-s + 0.710i·13-s − 7.65i·17-s − 1.02·19-s − 6.36i·23-s + (−1.81 − 4.65i)25-s − 0.234·29-s − 6.88·31-s + (−5.77 − 3.94i)35-s − 8.84i·37-s + 11.0·41-s − 6.28i·43-s − 3.15i·47-s + ⋯
L(s)  = 1  + (−0.564 + 0.825i)5-s + 1.18i·7-s + 0.301·11-s + 0.196i·13-s − 1.85i·17-s − 0.235·19-s − 1.32i·23-s + (−0.363 − 0.931i)25-s − 0.0434·29-s − 1.23·31-s + (−0.976 − 0.667i)35-s − 1.45i·37-s + 1.72·41-s − 0.957i·43-s − 0.459i·47-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((0.825+0.564i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s+1/2) L(s)=((0.825+0.564i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.825+0.564i0.825 + 0.564i
Analytic conductor: 31.620731.6207
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3960(3169,⋅)\chi_{3960} (3169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :1/2), 0.825+0.564i)(2,\ 3960,\ (\ :1/2),\ 0.825 + 0.564i)

Particular Values

L(1)L(1) ≈\approx 1.2540995221.254099522
L(12)L(\frac12) ≈\approx 1.2540995221.254099522
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 1
5 1+(1.26−1.84i)T 1 + (1.26 - 1.84i)T
11 1−T 1 - T
good7 1−3.12iT−7T2 1 - 3.12iT - 7T^{2}
13 1−0.710iT−13T2 1 - 0.710iT - 13T^{2}
17 1+7.65iT−17T2 1 + 7.65iT - 17T^{2}
19 1+1.02T+19T2 1 + 1.02T + 19T^{2}
23 1+6.36iT−23T2 1 + 6.36iT - 23T^{2}
29 1+0.234T+29T2 1 + 0.234T + 29T^{2}
31 1+6.88T+31T2 1 + 6.88T + 31T^{2}
37 1+8.84iT−37T2 1 + 8.84iT - 37T^{2}
41 1−11.0T+41T2 1 - 11.0T + 41T^{2}
43 1+6.28iT−43T2 1 + 6.28iT - 43T^{2}
47 1+3.15iT−47T2 1 + 3.15iT - 47T^{2}
53 1−8.79iT−53T2 1 - 8.79iT - 53T^{2}
59 1−8.01T+59T2 1 - 8.01T + 59T^{2}
61 1+10.1T+61T2 1 + 10.1T + 61T^{2}
67 1−4.21iT−67T2 1 - 4.21iT - 67T^{2}
71 1−10.5T+71T2 1 - 10.5T + 71T^{2}
73 1−5.47iT−73T2 1 - 5.47iT - 73T^{2}
79 1−1.50T+79T2 1 - 1.50T + 79T^{2}
83 1+8.70iT−83T2 1 + 8.70iT - 83T^{2}
89 1−4.64T+89T2 1 - 4.64T + 89T^{2}
97 1+17.8iT−97T2 1 + 17.8iT - 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

−8.500862104030783966189806008645, −7.43741939954125223515232345225, −7.09826667229669765028570438034, −6.16095256688566250823955591289, −5.50788666211757093467209597230, −4.56654254460117817164858187147, −3.72963928759149152589977376434, −2.67757905296130804655499703046, −2.26541971433270581174153569977, −0.43183182134641178541071802495, 0.982647284149790886781367879995, 1.76091786270180830376772280644, 3.42625216275978042920187260918, 3.91833009551222517443305690062, 4.59006632553667391901618487123, 5.53319187594069280193363213036, 6.32495464924314219399901250655, 7.20961304963735295382578053611, 7.914103497649092749273109546709, 8.303796876443072630377583720460

Graph of the ZZ-function along the critical line