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Properties

Label 2-363-33.20-c0-0-0
Degree $2$
Conductor $363$
Sign $0.569 + 0.821i$
Analytic cond. $0.181160$
Root an. cond. $0.425629$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)9-s − 12-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (−0.618 + 1.90i)31-s + (−0.809 + 0.587i)36-s + (1.61 + 1.17i)37-s + (0.809 + 0.587i)48-s + (−0.309 − 0.951i)49-s + (0.309 − 0.951i)64-s − 2·67-s + (−0.309 + 0.951i)75-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)9-s − 12-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (−0.618 + 1.90i)31-s + (−0.809 + 0.587i)36-s + (1.61 + 1.17i)37-s + (0.809 + 0.587i)48-s + (−0.309 − 0.951i)49-s + (0.309 − 0.951i)64-s − 2·67-s + (−0.309 + 0.951i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(0.181160\)
Root analytic conductor: \(0.425629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :0),\ 0.569 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8787395909\)
\(L(\frac12)\) \(\approx\) \(0.8787395909\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.60334923423243049189003682351, −10.35947963128729202997002505678, −9.525875442102522616499636580942, −8.769073372381244744378607124484, −7.925887670773080368600918124268, −6.80864664172881355213017516475, −5.71398631229508462563979728516, −4.45532730895249826227240071082, −3.23112915873196011125024679814, −1.54848778935769911800562078805, 2.51911550497708430071669515855, 3.81368417415172572109019823283, 4.51339613770582073786590703460, 5.81687527139242650830957661593, 7.55258136841995786417402856092, 8.068266425726644965230629538474, 9.189110537970495911476189432514, 9.612937094737320800479932744133, 10.73428730324520278723532565357, 11.82127054859491686278185255043

Graph of the $Z$-function along the critical line