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Properties

Label 2-363-11.3-c1-0-4
Degree 22
Conductor 363363
Sign 0.782−0.622i0.782 - 0.622i
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  + (0.809 − 0.587i)2-s + (−0.309 − 0.951i)3-s + (−0.309 + 0.951i)4-s + (1.61 + 1.17i)5-s + (−0.809 − 0.587i)6-s + (−1.23 + 3.80i)7-s + (0.927 + 2.85i)8-s + (−0.809 + 0.587i)9-s + 2·10-s + 0.999·12-s + (−1.61 + 1.17i)13-s + (1.23 + 3.80i)14-s + (0.618 − 1.90i)15-s + (0.809 + 0.587i)16-s + (−1.61 − 1.17i)17-s + (−0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.178 − 0.549i)3-s + (−0.154 + 0.475i)4-s + (0.723 + 0.525i)5-s + (−0.330 − 0.239i)6-s + (−0.467 + 1.43i)7-s + (0.327 + 1.00i)8-s + (−0.269 + 0.195i)9-s + 0.632·10-s + 0.288·12-s + (−0.448 + 0.326i)13-s + (0.330 + 1.01i)14-s + (0.159 − 0.491i)15-s + (0.202 + 0.146i)16-s + (−0.392 − 0.285i)17-s + (−0.0728 + 0.224i)18-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((0.782−0.622i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((0.782−0.622i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: 0.782−0.622i0.782 - 0.622i
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ363(124,⋅)\chi_{363} (124, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 363, ( :1/2), 0.782−0.622i)(2,\ 363,\ (\ :1/2),\ 0.782 - 0.622i)

Particular Values

L(1)L(1) ≈\approx 1.56244+0.545927i1.56244 + 0.545927i
L(12)L(\frac12) ≈\approx 1.56244+0.545927i1.56244 + 0.545927i
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+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
11 1 1
good2 1+(−0.809+0.587i)T+(0.618−1.90i)T2 1 + (-0.809 + 0.587i)T + (0.618 - 1.90i)T^{2}
5 1+(−1.61−1.17i)T+(1.54+4.75i)T2 1 + (-1.61 - 1.17i)T + (1.54 + 4.75i)T^{2}
7 1+(1.23−3.80i)T+(−5.66−4.11i)T2 1 + (1.23 - 3.80i)T + (-5.66 - 4.11i)T^{2}
13 1+(1.61−1.17i)T+(4.01−12.3i)T2 1 + (1.61 - 1.17i)T + (4.01 - 12.3i)T^{2}
17 1+(1.61+1.17i)T+(5.25+16.1i)T2 1 + (1.61 + 1.17i)T + (5.25 + 16.1i)T^{2}
19 1+(−15.3+11.1i)T2 1 + (-15.3 + 11.1i)T^{2}
23 1−8T+23T2 1 - 8T + 23T^{2}
29 1+(−1.85+5.70i)T+(−23.4−17.0i)T2 1 + (-1.85 + 5.70i)T + (-23.4 - 17.0i)T^{2}
31 1+(−6.47+4.70i)T+(9.57−29.4i)T2 1 + (-6.47 + 4.70i)T + (9.57 - 29.4i)T^{2}
37 1+(−1.85+5.70i)T+(−29.9−21.7i)T2 1 + (-1.85 + 5.70i)T + (-29.9 - 21.7i)T^{2}
41 1+(−0.618−1.90i)T+(−33.1+24.0i)T2 1 + (-0.618 - 1.90i)T + (-33.1 + 24.0i)T^{2}
43 1+43T2 1 + 43T^{2}
47 1+(−2.47−7.60i)T+(−38.0+27.6i)T2 1 + (-2.47 - 7.60i)T + (-38.0 + 27.6i)T^{2}
53 1+(4.85−3.52i)T+(16.3−50.4i)T2 1 + (4.85 - 3.52i)T + (16.3 - 50.4i)T^{2}
59 1+(1.23−3.80i)T+(−47.7−34.6i)T2 1 + (1.23 - 3.80i)T + (-47.7 - 34.6i)T^{2}
61 1+(−4.85−3.52i)T+(18.8+58.0i)T2 1 + (-4.85 - 3.52i)T + (18.8 + 58.0i)T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1+(21.9+67.5i)T2 1 + (21.9 + 67.5i)T^{2}
73 1+(−4.32+13.3i)T+(−59.0−42.9i)T2 1 + (-4.32 + 13.3i)T + (-59.0 - 42.9i)T^{2}
79 1+(3.23−2.35i)T+(24.4−75.1i)T2 1 + (3.23 - 2.35i)T + (24.4 - 75.1i)T^{2}
83 1+(−9.70−7.05i)T+(25.6+78.9i)T2 1 + (-9.70 - 7.05i)T + (25.6 + 78.9i)T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+(1.61−1.17i)T+(29.9−92.2i)T2 1 + (1.61 - 1.17i)T + (29.9 - 92.2i)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

−11.77031720719592189043859172571, −10.95081130118889611654642108973, −9.596710929943466167605038587549, −8.857557702407961576245528068263, −7.73087969932428976566360110184, −6.52641738903461047931731754318, −5.72379071659773150020689052389, −4.59680674801099632065148050062, −2.82018581473423530918529981518, −2.36026984055612069969970144140, 1.04179043386401459710073506741, 3.41392332888437310674738873242, 4.64689388108015808252163385028, 5.21566021857320539991304636464, 6.45536507269735971089444455845, 7.14275722682849799658408568502, 8.741389982839601996509955212134, 9.751132638106307679571016945129, 10.26096586068518123325551155818, 11.04374390321835708829616362653

Graph of the ZZ-function along the critical line