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Properties

Label 2-363-11.9-c1-0-14
Degree 22
Conductor 363363
Sign −0.577+0.816i-0.577 + 0.816i
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  + (0.535 − 1.64i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.927 − 2.85i)5-s + (−0.535 − 1.64i)6-s + (2.80 + 2.03i)7-s + (1.40 − 1.01i)8-s + (0.309 − 0.951i)9-s − 5.19·10-s − 0.999·12-s + (−0.535 + 1.64i)13-s + (4.85 − 3.52i)14-s + (−2.42 − 1.76i)15-s + (−1.54 − 4.75i)16-s + (0.535 + 1.64i)17-s + (−1.40 − 1.01i)18-s + ⋯
L(s)  = 1  + (0.378 − 1.16i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.414 − 1.27i)5-s + (−0.218 − 0.672i)6-s + (1.05 + 0.769i)7-s + (0.495 − 0.359i)8-s + (0.103 − 0.317i)9-s − 1.64·10-s − 0.288·12-s + (−0.148 + 0.456i)13-s + (1.29 − 0.942i)14-s + (−0.626 − 0.455i)15-s + (−0.386 − 1.18i)16-s + (0.129 + 0.399i)17-s + (−0.330 − 0.239i)18-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((−0.577+0.816i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((−0.577+0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: −0.577+0.816i-0.577 + 0.816i
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), −0.577+0.816i)(2,\ 363,\ (\ :1/2),\ -0.577 + 0.816i)

Particular Values

L(1)L(1) ≈\approx 0.926172−1.78874i0.926172 - 1.78874i
L(12)L(\frac12) ≈\approx 0.926172−1.78874i0.926172 - 1.78874i
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+(−0.535+1.64i)T+(−1.61−1.17i)T2 1 + (-0.535 + 1.64i)T + (-1.61 - 1.17i)T^{2}
5 1+(0.927+2.85i)T+(−4.04+2.93i)T2 1 + (0.927 + 2.85i)T + (-4.04 + 2.93i)T^{2}
7 1+(−2.80−2.03i)T+(2.16+6.65i)T2 1 + (-2.80 - 2.03i)T + (2.16 + 6.65i)T^{2}
13 1+(0.535−1.64i)T+(−10.5−7.64i)T2 1 + (0.535 - 1.64i)T + (-10.5 - 7.64i)T^{2}
17 1+(−0.535−1.64i)T+(−13.7+9.99i)T2 1 + (-0.535 - 1.64i)T + (-13.7 + 9.99i)T^{2}
19 1+(5.60−4.07i)T+(5.87−18.0i)T2 1 + (5.60 - 4.07i)T + (5.87 - 18.0i)T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1+(−1.40−1.01i)T+(8.96+27.5i)T2 1 + (-1.40 - 1.01i)T + (8.96 + 27.5i)T^{2}
31 1+(−1.23+3.80i)T+(−25.0−18.2i)T2 1 + (-1.23 + 3.80i)T + (-25.0 - 18.2i)T^{2}
37 1+(−8.89−6.46i)T+(11.4+35.1i)T2 1 + (-8.89 - 6.46i)T + (11.4 + 35.1i)T^{2}
41 1+(−1.40+1.01i)T+(12.6−38.9i)T2 1 + (-1.40 + 1.01i)T + (12.6 - 38.9i)T^{2}
43 1−3.46T+43T2 1 - 3.46T + 43T^{2}
47 1+(14.5−44.6i)T2 1 + (14.5 - 44.6i)T^{2}
53 1+(2.78−8.55i)T+(−42.8−31.1i)T2 1 + (2.78 - 8.55i)T + (-42.8 - 31.1i)T^{2}
59 1+(−4.85−3.52i)T+(18.2+56.1i)T2 1 + (-4.85 - 3.52i)T + (18.2 + 56.1i)T^{2}
61 1+(−49.3+35.8i)T2 1 + (-49.3 + 35.8i)T^{2}
67 1+2T+67T2 1 + 2T + 67T^{2}
71 1+(1.85+5.70i)T+(−57.4+41.7i)T2 1 + (1.85 + 5.70i)T + (-57.4 + 41.7i)T^{2}
73 1+(5.60+4.07i)T+(22.5+69.4i)T2 1 + (5.60 + 4.07i)T + (22.5 + 69.4i)T^{2}
79 1+(−63.9−46.4i)T2 1 + (-63.9 - 46.4i)T^{2}
83 1+(−67.1+48.7i)T2 1 + (-67.1 + 48.7i)T^{2}
89 1−9T+89T2 1 - 9T + 89T^{2}
97 1+(2.16−6.65i)T+(−78.4−57.0i)T2 1 + (2.16 - 6.65i)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

−11.49649722059092517567676050119, −10.36472215565711527994248399896, −9.255216130614050327366296639752, −8.319502316688747609015933021903, −7.80088092896954727409020780334, −6.04473797560603592611835847145, −4.64389207856629246177084570818, −4.05061497994884935023729049188, −2.35394230324454250808928624287, −1.43492699702706317149551454200, 2.40444172513942350548236196685, 3.97713371202421931921103275932, 4.80527792630616643529422370757, 6.15827007181814166542170597300, 7.14065158232685482207453373692, 7.72405983416698438555192785608, 8.493292356411723490582523298062, 10.08502891966320945913348904617, 10.87346241444923413655565456279, 11.37701627675583626168566167010

Graph of the ZZ-function along the critical line