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Properties

Label 2-476-476.135-c0-0-3
Degree 22
Conductor 476476
Sign −0.997+0.0633i-0.997 + 0.0633i
Analytic cond. 0.2375540.237554
Root an. cond. 0.4873960.487396
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  + (0.5 − 0.866i)2-s + (−0.866 − 1.5i)3-s + (−0.499 − 0.866i)4-s − 1.73·6-s − i·7-s − 0.999·8-s + (−1 + 1.73i)9-s + (0.866 + 1.5i)11-s + (−0.866 + 1.49i)12-s + 13-s + (−0.866 − 0.5i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (1 + 1.73i)18-s + (−1.5 + 0.866i)21-s + 1.73·22-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.866 − 1.5i)3-s + (−0.499 − 0.866i)4-s − 1.73·6-s − i·7-s − 0.999·8-s + (−1 + 1.73i)9-s + (0.866 + 1.5i)11-s + (−0.866 + 1.49i)12-s + 13-s + (−0.866 − 0.5i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (1 + 1.73i)18-s + (−1.5 + 0.866i)21-s + 1.73·22-s + ⋯

Functional equation

Λ(s)=(476s/2 ΓC(s) L(s)=((−0.997+0.0633i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(476s/2 ΓC(s) L(s)=((−0.997+0.0633i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 476476    =    22⋅7⋅172^{2} \cdot 7 \cdot 17
Sign: −0.997+0.0633i-0.997 + 0.0633i
Analytic conductor: 0.2375540.237554
Root analytic conductor: 0.4873960.487396
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ476(135,⋅)\chi_{476} (135, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 476, ( :0), −0.997+0.0633i)(2,\ 476,\ (\ :0),\ -0.997 + 0.0633i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.79664790760.7966479076
L(12)L(\frac12) ≈\approx 0.79664790760.7966479076
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+iT 1 + iT
17 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good3 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2}
13 1−T+T2 1 - T + T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−1.73T+T2 1 - 1.73T + T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)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

−11.09535139220431389658158850983, −10.26181528170142932020872155541, −9.189168278053678897071393153337, −7.79619662493473746985943868253, −6.81473999656473342397938340318, −6.28913316104520850601139058727, −4.97159460935988884613838754383, −3.97962228810159768289381329074, −2.17144802826572071518473476174, −1.08987278157440062378599310698, 3.39453644134763664225732626843, 4.01465038350638189169196351049, 5.27576068283942987298436104519, 5.91439906930685071175477783227, 6.47252090941402724136107794406, 8.395387336984686484425394313451, 8.855694864116422952925778850403, 9.699092736512341396188385167651, 11.13484282716842273688163895251, 11.36622786111648245464148500183

Graph of the ZZ-function along the critical line