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Properties

Label 2-3960-1320.659-c0-0-18
Degree 22
Conductor 39603960
Sign 0.845+0.533i0.845 + 0.533i
Analytic cond. 1.976291.97629
Root an. cond. 1.405801.40580
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 + 0.382i)2-s + (0.707 + 0.707i)4-s − 5-s − 1.84i·7-s + (0.382 + 0.923i)8-s + (−0.923 − 0.382i)10-s + i·11-s − 0.765i·13-s + (0.707 − 1.70i)14-s + i·16-s − 1.84i·17-s + (−0.707 − 0.707i)20-s + (−0.382 + 0.923i)22-s + 25-s + (0.292 − 0.707i)26-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)2-s + (0.707 + 0.707i)4-s − 5-s − 1.84i·7-s + (0.382 + 0.923i)8-s + (−0.923 − 0.382i)10-s + i·11-s − 0.765i·13-s + (0.707 − 1.70i)14-s + i·16-s − 1.84i·17-s + (−0.707 − 0.707i)20-s + (−0.382 + 0.923i)22-s + 25-s + (0.292 − 0.707i)26-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((0.845+0.533i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s) L(s)=((0.845+0.533i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.845+0.533i0.845 + 0.533i
Analytic conductor: 1.976291.97629
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3960(1979,⋅)\chi_{3960} (1979, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :0), 0.845+0.533i)(2,\ 3960,\ (\ :0),\ 0.845 + 0.533i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8396823261.839682326
L(12)L(\frac12) ≈\approx 1.8396823261.839682326
L(1)L(1) 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+(−0.923−0.382i)T 1 + (-0.923 - 0.382i)T
3 1 1
5 1+T 1 + T
11 1−iT 1 - iT
good7 1+1.84iT−T2 1 + 1.84iT - T^{2}
13 1+0.765iT−T2 1 + 0.765iT - T^{2}
17 1+1.84iT−T2 1 + 1.84iT - T^{2}
19 1−T2 1 - T^{2}
23 1−T2 1 - T^{2}
29 1−T2 1 - T^{2}
31 1+1.41iT−T2 1 + 1.41iT - T^{2}
37 1+T2 1 + T^{2}
41 1+T2 1 + T^{2}
43 1−1.84T+T2 1 - 1.84T + T^{2}
47 1−T2 1 - T^{2}
53 1−T2 1 - T^{2}
59 1+1.41iT−T2 1 + 1.41iT - T^{2}
61 1+T2 1 + T^{2}
67 1−T2 1 - T^{2}
71 1−1.41T+T2 1 - 1.41T + T^{2}
73 1+0.765T+T2 1 + 0.765T + T^{2}
79 1+T2 1 + T^{2}
83 1−0.765iT−T2 1 - 0.765iT - T^{2}
89 1−2iT−T2 1 - 2iT - T^{2}
97 1−T2 1 - T^{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

−7.970040048872992545297260953376, −7.69771716700147128206404912947, −7.12839350234580140368931098813, −6.63347789635341590365165903140, −5.36137014831954117511195953272, −4.61507421280942553760464819622, −4.13535180987609702278457395751, −3.41580680596221167512385927598, −2.47155267645868571596373524698, −0.78830010746630318381565432904, 1.50373093101132212485002946156, 2.53370050212074824476007081608, 3.30978309763761941757398367279, 4.04789876546875778193022633948, 4.89257260678206837159340069290, 5.81387920219256326230450732216, 6.13524658515420667460840370069, 7.05572809999574825003584363120, 8.098923719770247606101203761315, 8.712823327739130949727053216259

Graph of the ZZ-function along the critical line