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Properties

Label 2-960-120.107-c0-0-0
Degree 22
Conductor 960960
Sign 0.229−0.973i0.229 - 0.973i
Analytic cond. 0.4791020.479102
Root an. cond. 0.6921720.692172
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  + (−0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s + (−1 + i)7-s + 1.00i·9-s + 1.41i·11-s + 1.00i·15-s + 1.41·21-s + 1.00i·25-s + (0.707 − 0.707i)27-s + 1.41i·29-s + (1.00 − 1.00i)33-s + 1.41·35-s + (0.707 − 0.707i)45-s − i·49-s + (−1.41 + 1.41i)53-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s + (−1 + i)7-s + 1.00i·9-s + 1.41i·11-s + 1.00i·15-s + 1.41·21-s + 1.00i·25-s + (0.707 − 0.707i)27-s + 1.41i·29-s + (1.00 − 1.00i)33-s + 1.41·35-s + (0.707 − 0.707i)45-s − i·49-s + (−1.41 + 1.41i)53-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.229−0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s) L(s)=((0.229−0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.229−0.973i0.229 - 0.973i
Analytic conductor: 0.4791020.479102
Root analytic conductor: 0.6921720.692172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ960(287,⋅)\chi_{960} (287, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :0), 0.229−0.973i)(2,\ 960,\ (\ :0),\ 0.229 - 0.973i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.41216160810.4121616081
L(12)L(\frac12) ≈\approx 0.41216160810.4121616081
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 1
3 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
5 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good7 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
11 1−1.41iT−T2 1 - 1.41iT - T^{2}
13 1+iT2 1 + iT^{2}
17 1−iT2 1 - iT^{2}
19 1−T2 1 - T^{2}
23 1−iT2 1 - iT^{2}
29 1−1.41iT−T2 1 - 1.41iT - T^{2}
31 1−T2 1 - T^{2}
37 1−iT2 1 - iT^{2}
41 1−T2 1 - T^{2}
43 1−iT2 1 - iT^{2}
47 1+iT2 1 + iT^{2}
53 1+(1.41−1.41i)T−iT2 1 + (1.41 - 1.41i)T - iT^{2}
59 1+1.41T+T2 1 + 1.41T + T^{2}
61 1−T2 1 - T^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
79 1+T2 1 + T^{2}
83 1+iT2 1 + iT^{2}
89 1+T2 1 + T^{2}
97 1+(1+i)T+iT2 1 + (1 + i)T + iT^{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

−10.44823527644084265993014830073, −9.466643786857875089208930327978, −8.788144031158073430955137884716, −7.72860544172485611019105759924, −7.04614727082797716696462624221, −6.15564799367901701608401138345, −5.23037029087958850839305820986, −4.44035228393569833582695394968, −2.97122021121797226514391073316, −1.63671808758740279470374265084, 0.43406240891863601879244889421, 3.12961762722282852526811288830, 3.66559386123698103083288515228, 4.57888716449978539701770448898, 5.98651646807274834141218568921, 6.45328345462200126512332886374, 7.40413740577371857177523711192, 8.376009124828956151724252789596, 9.497752166396538145735674479316, 10.19048561541531564472396213365

Graph of the ZZ-function along the critical line