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Properties

Label 2-363-33.14-c0-0-0
Degree $2$
Conductor $363$
Sign $0.0694 + 0.997i$
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  + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)9-s − 0.999·12-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (1.61 − 1.17i)31-s + (0.309 + 0.951i)36-s + (−0.618 + 1.90i)37-s + (−0.309 + 0.951i)48-s + (0.809 + 0.587i)49-s + (−0.809 + 0.587i)64-s − 2·67-s + (0.809 − 0.587i)75-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)9-s − 0.999·12-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (1.61 − 1.17i)31-s + (0.309 + 0.951i)36-s + (−0.618 + 1.90i)37-s + (−0.309 + 0.951i)48-s + (0.809 + 0.587i)49-s + (−0.809 + 0.587i)64-s − 2·67-s + (0.809 − 0.587i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.0694 + 0.997i$
Analytic conductor: \(0.181160\)
Root analytic conductor: \(0.425629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :0),\ 0.0694 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7665710452\)
\(L(\frac12)\) \(\approx\) \(0.7665710452\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−11.49663922709092426005814638367, −10.64854228361270837765603463846, −9.725058391624146140220143299148, −8.568738621602930742608859516826, −7.47670976419178148087923270472, −6.56650594185616066295002626321, −5.80426373096249729933395900283, −4.75801903048621441368532879668, −2.73899493222222187686912804421, −1.34966010933792512130866683765, 2.69886119450899712520019757737, 3.81831537282911739701181412907, 4.78938295323439278206713657386, 6.08849349435628478958587913532, 7.12695092971005917993398219619, 8.344628753986667870938261585630, 9.011990249448143678941660543893, 10.20999832005157148837151867275, 10.90243723898159131067407603221, 11.92301488068478685493573834395

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