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Properties

Label 2-363-11.4-c1-0-10
Degree 22
Conductor 363363
Sign −0.659+0.751i-0.659 + 0.751i
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  + (−1.40 − 1.01i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (2.42 − 1.76i)5-s + (1.40 − 1.01i)6-s + (−1.07 − 3.29i)7-s + (−0.535 + 1.64i)8-s + (−0.809 − 0.587i)9-s − 5.19·10-s − 0.999·12-s + (1.40 + 1.01i)13-s + (−1.85 + 5.70i)14-s + (0.927 + 2.85i)15-s + (4.04 − 2.93i)16-s + (−1.40 + 1.01i)17-s + (0.535 + 1.64i)18-s + ⋯
L(s)  = 1  + (−0.990 − 0.719i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (1.08 − 0.788i)5-s + (0.572 − 0.415i)6-s + (−0.404 − 1.24i)7-s + (−0.189 + 0.582i)8-s + (−0.269 − 0.195i)9-s − 1.64·10-s − 0.288·12-s + (0.388 + 0.282i)13-s + (−0.495 + 1.52i)14-s + (0.239 + 0.736i)15-s + (1.01 − 0.734i)16-s + (−0.339 + 0.246i)17-s + (0.126 + 0.388i)18-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((−0.659+0.751i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((−0.659+0.751i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: −0.659+0.751i-0.659 + 0.751i
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ363(202,⋅)\chi_{363} (202, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 363, ( :1/2), −0.659+0.751i)(2,\ 363,\ (\ :1/2),\ -0.659 + 0.751i)

Particular Values

L(1)L(1) ≈\approx 0.296269−0.653897i0.296269 - 0.653897i
L(12)L(\frac12) ≈\approx 0.296269−0.653897i0.296269 - 0.653897i
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+(1.40+1.01i)T+(0.618+1.90i)T2 1 + (1.40 + 1.01i)T + (0.618 + 1.90i)T^{2}
5 1+(−2.42+1.76i)T+(1.54−4.75i)T2 1 + (-2.42 + 1.76i)T + (1.54 - 4.75i)T^{2}
7 1+(1.07+3.29i)T+(−5.66+4.11i)T2 1 + (1.07 + 3.29i)T + (-5.66 + 4.11i)T^{2}
13 1+(−1.40−1.01i)T+(4.01+12.3i)T2 1 + (-1.40 - 1.01i)T + (4.01 + 12.3i)T^{2}
17 1+(1.40−1.01i)T+(5.25−16.1i)T2 1 + (1.40 - 1.01i)T + (5.25 - 16.1i)T^{2}
19 1+(−2.14+6.58i)T+(−15.3−11.1i)T2 1 + (-2.14 + 6.58i)T + (-15.3 - 11.1i)T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1+(0.535+1.64i)T+(−23.4+17.0i)T2 1 + (0.535 + 1.64i)T + (-23.4 + 17.0i)T^{2}
31 1+(3.23+2.35i)T+(9.57+29.4i)T2 1 + (3.23 + 2.35i)T + (9.57 + 29.4i)T^{2}
37 1+(3.39+10.4i)T+(−29.9+21.7i)T2 1 + (3.39 + 10.4i)T + (-29.9 + 21.7i)T^{2}
41 1+(0.535−1.64i)T+(−33.1−24.0i)T2 1 + (0.535 - 1.64i)T + (-33.1 - 24.0i)T^{2}
43 1−3.46T+43T2 1 - 3.46T + 43T^{2}
47 1+(−38.0−27.6i)T2 1 + (-38.0 - 27.6i)T^{2}
53 1+(−7.28−5.29i)T+(16.3+50.4i)T2 1 + (-7.28 - 5.29i)T + (16.3 + 50.4i)T^{2}
59 1+(1.85+5.70i)T+(−47.7+34.6i)T2 1 + (1.85 + 5.70i)T + (-47.7 + 34.6i)T^{2}
61 1+(18.8−58.0i)T2 1 + (18.8 - 58.0i)T^{2}
67 1+2T+67T2 1 + 2T + 67T^{2}
71 1+(−4.85+3.52i)T+(21.9−67.5i)T2 1 + (-4.85 + 3.52i)T + (21.9 - 67.5i)T^{2}
73 1+(−2.14−6.58i)T+(−59.0+42.9i)T2 1 + (-2.14 - 6.58i)T + (-59.0 + 42.9i)T^{2}
79 1+(24.4+75.1i)T2 1 + (24.4 + 75.1i)T^{2}
83 1+(25.6−78.9i)T2 1 + (25.6 - 78.9i)T^{2}
89 1−9T+89T2 1 - 9T + 89T^{2}
97 1+(−5.66−4.11i)T+(29.9+92.2i)T2 1 + (-5.66 - 4.11i)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

−10.75760871201764765517674130351, −10.14903837265714745238669008090, −9.327205882649217624252019710871, −8.922216903820059997046422527985, −7.55306187285961558761778222401, −6.17968794026763141748679282076, −5.13477699623572997110190246485, −3.88575524645190048200182066540, −2.14891460825547206488438341386, −0.69066536999235802947591352818, 1.88294261858518049186058766262, 3.26128540389374304852470714138, 5.68636494407671867570284717921, 6.11371113063223955329067002202, 6.99004298891475597601711863120, 8.085211006446370885746700496722, 8.887424843880581459253936569320, 9.825918203469499321583799647888, 10.40819516355744906395345353561, 11.84261289678431140258819798826

Graph of the ZZ-function along the critical line