Properties

Label 2-363-1.1-c1-0-14
Degree $2$
Conductor $363$
Sign $-1$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 3-s + 0.618·4-s − 0.381·5-s − 1.61·6-s − 3·7-s + 2.23·8-s + 9-s + 0.618·10-s + 0.618·12-s − 6.23·13-s + 4.85·14-s − 0.381·15-s − 4.85·16-s + 0.618·17-s − 1.61·18-s + 0.854·19-s − 0.236·20-s − 3·21-s − 5.47·23-s + 2.23·24-s − 4.85·25-s + 10.0·26-s + 27-s − 1.85·28-s + 4.47·29-s + 0.618·30-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.577·3-s + 0.309·4-s − 0.170·5-s − 0.660·6-s − 1.13·7-s + 0.790·8-s + 0.333·9-s + 0.195·10-s + 0.178·12-s − 1.72·13-s + 1.29·14-s − 0.0986·15-s − 1.21·16-s + 0.149·17-s − 0.381·18-s + 0.195·19-s − 0.0527·20-s − 0.654·21-s − 1.14·23-s + 0.456·24-s − 0.970·25-s + 1.97·26-s + 0.192·27-s − 0.350·28-s + 0.830·29-s + 0.112·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + 1.61T + 2T^{2} \)
5 \( 1 + 0.381T + 5T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
13 \( 1 + 6.23T + 13T^{2} \)
17 \( 1 - 0.618T + 17T^{2} \)
19 \( 1 - 0.854T + 19T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 3.85T + 31T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 + 5.94T + 41T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 + 0.618T + 47T^{2} \)
53 \( 1 + 7.38T + 53T^{2} \)
59 \( 1 + 5.32T + 59T^{2} \)
61 \( 1 + 1.14T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + 1.23T + 73T^{2} \)
79 \( 1 + 0.527T + 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 - 9.47T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−10.44962004892279487113516353833, −9.768570950460357282081378252608, −9.368161890918320204769588635155, −8.193649732012210798534917023753, −7.50470918349601455325913291569, −6.56497603865813640787210971213, −4.93286077599260726790850476019, −3.58014662113298503108212757707, −2.15407132052680465264893855070, 0, 2.15407132052680465264893855070, 3.58014662113298503108212757707, 4.93286077599260726790850476019, 6.56497603865813640787210971213, 7.50470918349601455325913291569, 8.193649732012210798534917023753, 9.368161890918320204769588635155, 9.768570950460357282081378252608, 10.44962004892279487113516353833

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