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Properties

Label 2-3960-5.4-c1-0-69
Degree 22
Conductor 39603960
Sign −0.894−0.447i-0.894 - 0.447i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 2i)5-s − 4i·7-s − 11-s − 6i·13-s + 2i·17-s − 4·19-s − 6i·23-s + (−3 − 4i)25-s − 2·29-s + 8·31-s + (8 + 4i)35-s + 8i·37-s − 6·41-s + 12i·43-s + 10i·47-s + ⋯
L(s)  = 1  + (−0.447 + 0.894i)5-s − 1.51i·7-s − 0.301·11-s − 1.66i·13-s + 0.485i·17-s − 0.917·19-s − 1.25i·23-s + (−0.600 − 0.800i)25-s − 0.371·29-s + 1.43·31-s + (1.35 + 0.676i)35-s + 1.31i·37-s − 0.937·41-s + 1.82i·43-s + 1.45i·47-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s+1/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: −0.894−0.447i-0.894 - 0.447i
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: 11
Selberg data: (2, 3960, ( :1/2), −0.894−0.447i)(2,\ 3960,\ (\ :1/2),\ -0.894 - 0.447i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+(1−2i)T 1 + (1 - 2i)T
11 1+T 1 + T
good7 1+4iT−7T2 1 + 4iT - 7T^{2}
13 1+6iT−13T2 1 + 6iT - 13T^{2}
17 1−2iT−17T2 1 - 2iT - 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1−8iT−37T2 1 - 8iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−12iT−43T2 1 - 12iT - 43T^{2}
47 1−10iT−47T2 1 - 10iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1+10T+61T2 1 + 10T + 61T^{2}
67 1+2iT−67T2 1 + 2iT - 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1−2iT−73T2 1 - 2iT - 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1−4iT−83T2 1 - 4iT - 83T^{2}
89 1+14T+89T2 1 + 14T + 89T^{2}
97 1−4iT−97T2 1 - 4iT - 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

−8.094005497106503424522686677155, −7.38423699253185673021647901859, −6.52940496700225220335212698709, −6.12650605190121258613463840327, −4.76669528937552966059545611232, −4.24087541192893065647347719401, −3.25374408795945736250518516221, −2.71120007214133344256004293538, −1.14268767624092829376826341943, 0, 1.69763026056002695589171623761, 2.34095725083537300450962399157, 3.58617630609164311183359368232, 4.41388260512417558592231944777, 5.18471647514324138086839513350, 5.76071479032182738164383061399, 6.67664847513532635198327944940, 7.46098434412460710736999700070, 8.390077121471355053278739533051

Graph of the ZZ-function along the critical line