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Properties

Label 2-363-11.9-c1-0-8
Degree 22
Conductor 363363
Sign 0.944+0.329i0.944 + 0.329i
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  + (0.190 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (1.30 + 0.951i)4-s + (−0.809 − 2.48i)5-s + (0.190 + 0.587i)6-s + (2.42 + 1.76i)7-s + (1.80 − 1.31i)8-s + (0.309 − 0.951i)9-s − 1.61·10-s − 1.61·12-s + (−0.545 + 1.67i)13-s + (1.5 − 1.08i)14-s + (2.11 + 1.53i)15-s + (0.572 + 1.76i)16-s + (−0.5 − 1.53i)17-s + (−0.5 − 0.363i)18-s + ⋯
L(s)  = 1  + (0.135 − 0.415i)2-s + (−0.467 + 0.339i)3-s + (0.654 + 0.475i)4-s + (−0.361 − 1.11i)5-s + (0.0779 + 0.239i)6-s + (0.917 + 0.666i)7-s + (0.639 − 0.464i)8-s + (0.103 − 0.317i)9-s − 0.511·10-s − 0.467·12-s + (−0.151 + 0.465i)13-s + (0.400 − 0.291i)14-s + (0.546 + 0.397i)15-s + (0.143 + 0.440i)16-s + (−0.121 − 0.373i)17-s + (−0.117 − 0.0856i)18-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((0.944+0.329i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((0.944+0.329i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: 0.944+0.329i0.944 + 0.329i
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ363(130,⋅)\chi_{363} (130, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 363, ( :1/2), 0.944+0.329i)(2,\ 363,\ (\ :1/2),\ 0.944 + 0.329i)

Particular Values

L(1)L(1) ≈\approx 1.53544−0.260524i1.53544 - 0.260524i
L(12)L(\frac12) ≈\approx 1.53544−0.260524i1.53544 - 0.260524i
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+(−0.190+0.587i)T+(−1.61−1.17i)T2 1 + (-0.190 + 0.587i)T + (-1.61 - 1.17i)T^{2}
5 1+(0.809+2.48i)T+(−4.04+2.93i)T2 1 + (0.809 + 2.48i)T + (-4.04 + 2.93i)T^{2}
7 1+(−2.42−1.76i)T+(2.16+6.65i)T2 1 + (-2.42 - 1.76i)T + (2.16 + 6.65i)T^{2}
13 1+(0.545−1.67i)T+(−10.5−7.64i)T2 1 + (0.545 - 1.67i)T + (-10.5 - 7.64i)T^{2}
17 1+(0.5+1.53i)T+(−13.7+9.99i)T2 1 + (0.5 + 1.53i)T + (-13.7 + 9.99i)T^{2}
19 1+(−4.73+3.44i)T+(5.87−18.0i)T2 1 + (-4.73 + 3.44i)T + (5.87 - 18.0i)T^{2}
23 1−3.47T+23T2 1 - 3.47T + 23T^{2}
29 1+(−3.61−2.62i)T+(8.96+27.5i)T2 1 + (-3.61 - 2.62i)T + (8.96 + 27.5i)T^{2}
31 1+(−0.881+2.71i)T+(−25.0−18.2i)T2 1 + (-0.881 + 2.71i)T + (-25.0 - 18.2i)T^{2}
37 1+(0.190+0.138i)T+(11.4+35.1i)T2 1 + (0.190 + 0.138i)T + (11.4 + 35.1i)T^{2}
41 1+(9.66−7.02i)T+(12.6−38.9i)T2 1 + (9.66 - 7.02i)T + (12.6 - 38.9i)T^{2}
43 1+6.23T+43T2 1 + 6.23T + 43T^{2}
47 1+(1.30−0.951i)T+(14.5−44.6i)T2 1 + (1.30 - 0.951i)T + (14.5 - 44.6i)T^{2}
53 1+(2.97−9.14i)T+(−42.8−31.1i)T2 1 + (2.97 - 9.14i)T + (-42.8 - 31.1i)T^{2}
59 1+(8.35+6.06i)T+(18.2+56.1i)T2 1 + (8.35 + 6.06i)T + (18.2 + 56.1i)T^{2}
61 1+(2.42+7.46i)T+(−49.3+35.8i)T2 1 + (2.42 + 7.46i)T + (-49.3 + 35.8i)T^{2}
67 1+9.56T+67T2 1 + 9.56T + 67T^{2}
71 1+(1.71+5.29i)T+(−57.4+41.7i)T2 1 + (1.71 + 5.29i)T + (-57.4 + 41.7i)T^{2}
73 1+(2.61+1.90i)T+(22.5+69.4i)T2 1 + (2.61 + 1.90i)T + (22.5 + 69.4i)T^{2}
79 1+(2.92−9.00i)T+(−63.9−46.4i)T2 1 + (2.92 - 9.00i)T + (-63.9 - 46.4i)T^{2}
83 1+(0.218+0.673i)T+(−67.1+48.7i)T2 1 + (0.218 + 0.673i)T + (-67.1 + 48.7i)T^{2}
89 1−0.527T+89T2 1 - 0.527T + 89T^{2}
97 1+(4.33−13.3i)T+(−78.4−57.0i)T2 1 + (4.33 - 13.3i)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

−11.68234759032165547880557836736, −10.80890217384698239170143399636, −9.513340246454958810914141499282, −8.629486171163322095248239057448, −7.72623546436009186779480678521, −6.59401636727049831138710924878, −5.07337895770830217275557492218, −4.60851830779525113903064995186, −3.05223612619032627501002467731, −1.43607979429484885392370960091, 1.51756182927486248169965921219, 3.14997464555285877804653587573, 4.77994006720615815586308854130, 5.79045300097549793679347253391, 6.88360530153483198766165017126, 7.37859679750734524258532826405, 8.235321082445680556134545230173, 10.18552226632240301940468707306, 10.58556190447344945042600258684, 11.42625515564436929233262615982

Graph of the ZZ-function along the critical line