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Properties

Label 2-363-11.5-c1-0-5
Degree 22
Conductor 363363
Sign −0.353−0.935i-0.353 - 0.935i
Analytic cond. 2.898562.89856
Root an. cond. 1.702511.70251
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (0.809 + 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.618 + 1.90i)5-s + (−0.309 + 0.951i)6-s + (−3.23 + 2.35i)7-s + (2.42 + 1.76i)8-s + (0.309 + 0.951i)9-s − 1.99·10-s + 12-s + (−0.618 − 1.90i)13-s + (−3.23 − 2.35i)14-s + (−1.61 + 1.17i)15-s + (−0.309 + 0.951i)16-s + (−0.618 + 1.90i)17-s + (−0.809 + 0.587i)18-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.467 + 0.339i)3-s + (0.404 − 0.293i)4-s + (−0.276 + 0.850i)5-s + (−0.126 + 0.388i)6-s + (−1.22 + 0.888i)7-s + (0.858 + 0.623i)8-s + (0.103 + 0.317i)9-s − 0.632·10-s + 0.288·12-s + (−0.171 − 0.527i)13-s + (−0.864 − 0.628i)14-s + (−0.417 + 0.303i)15-s + (−0.0772 + 0.237i)16-s + (−0.149 + 0.461i)17-s + (−0.190 + 0.138i)18-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((−0.353−0.935i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((−0.353−0.935i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: −0.353−0.935i-0.353 - 0.935i
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ363(148,⋅)\chi_{363} (148, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 363, ( :1/2), −0.353−0.935i)(2,\ 363,\ (\ :1/2),\ -0.353 - 0.935i)

Particular Values

L(1)L(1) ≈\approx 0.974455+1.41055i0.974455 + 1.41055i
L(12)L(\frac12) ≈\approx 0.974455+1.41055i0.974455 + 1.41055i
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)
bad3 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
11 1 1
good2 1+(−0.309−0.951i)T+(−1.61+1.17i)T2 1 + (-0.309 - 0.951i)T + (-1.61 + 1.17i)T^{2}
5 1+(0.618−1.90i)T+(−4.04−2.93i)T2 1 + (0.618 - 1.90i)T + (-4.04 - 2.93i)T^{2}
7 1+(3.23−2.35i)T+(2.16−6.65i)T2 1 + (3.23 - 2.35i)T + (2.16 - 6.65i)T^{2}
13 1+(0.618+1.90i)T+(−10.5+7.64i)T2 1 + (0.618 + 1.90i)T + (-10.5 + 7.64i)T^{2}
17 1+(0.618−1.90i)T+(−13.7−9.99i)T2 1 + (0.618 - 1.90i)T + (-13.7 - 9.99i)T^{2}
19 1+(5.87+18.0i)T2 1 + (5.87 + 18.0i)T^{2}
23 1−8T+23T2 1 - 8T + 23T^{2}
29 1+(−4.85+3.52i)T+(8.96−27.5i)T2 1 + (-4.85 + 3.52i)T + (8.96 - 27.5i)T^{2}
31 1+(2.47+7.60i)T+(−25.0+18.2i)T2 1 + (2.47 + 7.60i)T + (-25.0 + 18.2i)T^{2}
37 1+(4.85−3.52i)T+(11.4−35.1i)T2 1 + (4.85 - 3.52i)T + (11.4 - 35.1i)T^{2}
41 1+(−1.61−1.17i)T+(12.6+38.9i)T2 1 + (-1.61 - 1.17i)T + (12.6 + 38.9i)T^{2}
43 1+43T2 1 + 43T^{2}
47 1+(6.47+4.70i)T+(14.5+44.6i)T2 1 + (6.47 + 4.70i)T + (14.5 + 44.6i)T^{2}
53 1+(−1.85−5.70i)T+(−42.8+31.1i)T2 1 + (-1.85 - 5.70i)T + (-42.8 + 31.1i)T^{2}
59 1+(−3.23+2.35i)T+(18.2−56.1i)T2 1 + (-3.23 + 2.35i)T + (18.2 - 56.1i)T^{2}
61 1+(−1.85+5.70i)T+(−49.3−35.8i)T2 1 + (-1.85 + 5.70i)T + (-49.3 - 35.8i)T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1+(−57.4−41.7i)T2 1 + (-57.4 - 41.7i)T^{2}
73 1+(−11.3+8.22i)T+(22.5−69.4i)T2 1 + (-11.3 + 8.22i)T + (22.5 - 69.4i)T^{2}
79 1+(1.23+3.80i)T+(−63.9+46.4i)T2 1 + (1.23 + 3.80i)T + (-63.9 + 46.4i)T^{2}
83 1+(−3.70+11.4i)T+(−67.1−48.7i)T2 1 + (-3.70 + 11.4i)T + (-67.1 - 48.7i)T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+(−0.618−1.90i)T+(−78.4+57.0i)T2 1 + (-0.618 - 1.90i)T + (-78.4 + 57.0i)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

−11.57778503536895069531849954518, −10.68701959529235408724984463823, −9.915337920478613587978758216247, −8.906440147328885491272228670600, −7.78122843560863100468635410167, −6.81057888190689486542293042825, −6.12939827787071646739158165392, −5.01996649427904164347262499036, −3.36630142253116666738903155914, −2.49305920810547584736436509893, 1.11815953635356804128445641742, 2.84159272493449859583913090954, 3.73277125721217829323215573586, 4.86006298046175427579273892712, 6.85404144959143447746038206651, 7.05653010587958517994736824559, 8.460933933069173747466104717324, 9.341752160223365752200375313903, 10.33201509040870328238300366019, 11.21859219303426925048665235909

Graph of the ZZ-function along the critical line