Properties

Label 2-363-1.1-c1-0-13
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  − 0.381·2-s − 3-s − 1.85·4-s + 1.61·5-s + 0.381·6-s − 7-s + 1.47·8-s + 9-s − 0.618·10-s + 1.85·12-s − 4.23·13-s + 0.381·14-s − 1.61·15-s + 3.14·16-s − 7.85·17-s − 0.381·18-s − 0.854·19-s − 3·20-s + 21-s − 4.23·23-s − 1.47·24-s − 2.38·25-s + 1.61·26-s − 27-s + 1.85·28-s − 6·29-s + 0.618·30-s + ⋯
L(s)  = 1  − 0.270·2-s − 0.577·3-s − 0.927·4-s + 0.723·5-s + 0.155·6-s − 0.377·7-s + 0.520·8-s + 0.333·9-s − 0.195·10-s + 0.535·12-s − 1.17·13-s + 0.102·14-s − 0.417·15-s + 0.786·16-s − 1.90·17-s − 0.0900·18-s − 0.195·19-s − 0.670·20-s + 0.218·21-s − 0.883·23-s − 0.300·24-s − 0.476·25-s + 0.317·26-s − 0.192·27-s + 0.350·28-s − 1.11·29-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 \( 1 + 0.381T + 2T^{2} \)
5 \( 1 - 1.61T + 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
13 \( 1 + 4.23T + 13T^{2} \)
17 \( 1 + 7.85T + 17T^{2} \)
19 \( 1 + 0.854T + 19T^{2} \)
23 \( 1 + 4.23T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 + 1.76T + 37T^{2} \)
41 \( 1 + 4.23T + 41T^{2} \)
43 \( 1 - 6.70T + 43T^{2} \)
47 \( 1 - 1.09T + 47T^{2} \)
53 \( 1 + 2.61T + 53T^{2} \)
59 \( 1 - 9.61T + 59T^{2} \)
61 \( 1 - 8.56T + 61T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 + 5.32T + 71T^{2} \)
73 \( 1 - 7.70T + 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 7.47T + 83T^{2} \)
89 \( 1 + 3.76T + 89T^{2} \)
97 \( 1 - 1.14T + 97T^{2} \)
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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

−10.78289067142785881077653490406, −9.863709486678904235572054203793, −9.396881642356561666798327339601, −8.313693641954583186886394218883, −7.08970735101694332655335795573, −6.05540157654284710868464828254, −5.03013832220991572063030127937, −4.08942123162730063370882791638, −2.14593612553279308138501529502, 0, 2.14593612553279308138501529502, 4.08942123162730063370882791638, 5.03013832220991572063030127937, 6.05540157654284710868464828254, 7.08970735101694332655335795573, 8.313693641954583186886394218883, 9.396881642356561666798327339601, 9.863709486678904235572054203793, 10.78289067142785881077653490406

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