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Properties

Label 2-812-812.447-c0-0-5
Degree 22
Conductor 812812
Sign −0.560+0.828i-0.560 + 0.828i
Analytic cond. 0.4052400.405240
Root an. cond. 0.6365850.636585
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)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + 5-s − 1.00·6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s − 13-s + 1.00i·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (1 − i)17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + 5-s − 1.00·6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s − 13-s + 1.00i·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (1 − i)17-s + ⋯

Functional equation

Λ(s)=(812s/2 ΓC(s) L(s)=((−0.560+0.828i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(812s/2 ΓC(s) L(s)=((−0.560+0.828i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 812812    =    22⋅7⋅292^{2} \cdot 7 \cdot 29
Sign: −0.560+0.828i-0.560 + 0.828i
Analytic conductor: 0.4052400.405240
Root analytic conductor: 0.6365850.636585
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ812(447,⋅)\chi_{812} (447, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 812, ( :0), −0.560+0.828i)(2,\ 812,\ (\ :0),\ -0.560 + 0.828i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1858608461.185860846
L(12)L(\frac12) ≈\approx 1.1858608461.185860846
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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
7 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
29 1−iT 1 - iT
good3 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
5 1−T+T2 1 - T + T^{2}
11 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
13 1+T+T2 1 + T + T^{2}
17 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
19 1+iT2 1 + iT^{2}
23 1−T2 1 - T^{2}
31 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
37 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
41 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
43 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
47 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
53 1+T+T2 1 + T + T^{2}
59 1−1.41T+T2 1 - 1.41T + T^{2}
61 1−iT2 1 - iT^{2}
67 1−1.41T+T2 1 - 1.41T + T^{2}
71 1+T2 1 + T^{2}
73 1+iT2 1 + iT^{2}
79 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
83 1+1.41T+T2 1 + 1.41T + T^{2}
89 1−iT2 1 - iT^{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

−10.09681437149126920971043608558, −9.622521580747363288553250416430, −8.825992507029401808916993490060, −7.06936210023868316045297196296, −6.43384101726823293800780168390, −5.66457160963702274612591847344, −5.12212352678925465463818116964, −3.44528831791808685943812223509, −2.50223217345007271955091930500, −1.16408216204195686297358452815, 2.31772335607991590560388742198, 3.85970809793892810819533966428, 4.50703397517972438636839353799, 5.63344496041064564280497609817, 6.07358667201616527628982980510, 7.10103021722078039993648904651, 7.88662445369547412769383547768, 9.372490254713654281028812051233, 9.861070769290359387387494700874, 10.57855653999425758425935012046

Graph of the ZZ-function along the critical line