Properties

Label 2-297-1.1-c1-0-6
Degree $2$
Conductor $297$
Sign $-1$
Analytic cond. $2.37155$
Root an. cond. $1.53998$
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  − 2.73·2-s + 5.46·4-s + 0.732·5-s − 3.73·7-s − 9.46·8-s − 2·10-s − 11-s − 0.267·13-s + 10.1·14-s + 14.9·16-s + 4.19·17-s − 5.19·19-s + 4·20-s + 2.73·22-s − 8·23-s − 4.46·25-s + 0.732·26-s − 20.3·28-s + 1.26·29-s + 0.535·31-s − 21.8·32-s − 11.4·34-s − 2.73·35-s − 6.46·37-s + 14.1·38-s − 6.92·40-s − 1.46·41-s + ⋯
L(s)  = 1  − 1.93·2-s + 2.73·4-s + 0.327·5-s − 1.41·7-s − 3.34·8-s − 0.632·10-s − 0.301·11-s − 0.0743·13-s + 2.72·14-s + 3.73·16-s + 1.01·17-s − 1.19·19-s + 0.894·20-s + 0.582·22-s − 1.66·23-s − 0.892·25-s + 0.143·26-s − 3.85·28-s + 0.235·29-s + 0.0962·31-s − 3.86·32-s − 1.96·34-s − 0.461·35-s − 1.06·37-s + 2.30·38-s − 1.09·40-s − 0.228·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-1$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + 2.73T + 2T^{2} \)
5 \( 1 - 0.732T + 5T^{2} \)
7 \( 1 + 3.73T + 7T^{2} \)
13 \( 1 + 0.267T + 13T^{2} \)
17 \( 1 - 4.19T + 17T^{2} \)
19 \( 1 + 5.19T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 1.26T + 29T^{2} \)
31 \( 1 - 0.535T + 31T^{2} \)
37 \( 1 + 6.46T + 37T^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + 4.73T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 4.26T + 61T^{2} \)
67 \( 1 - 5.92T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 3.19T + 73T^{2} \)
79 \( 1 + 5.73T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 0.339T + 89T^{2} \)
97 \( 1 + 5.39T + 97T^{2} \)
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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

−10.82064368787251089401295184476, −9.922946070683342917394260083375, −9.699982788297855978122388060119, −8.511700238161327440662268732347, −7.69953954818979481840226331198, −6.55397274843630290556864924370, −5.94520275907457442358055304251, −3.35149336968103963333524756878, −2.02172474427898299129359430859, 0, 2.02172474427898299129359430859, 3.35149336968103963333524756878, 5.94520275907457442358055304251, 6.55397274843630290556864924370, 7.69953954818979481840226331198, 8.511700238161327440662268732347, 9.699982788297855978122388060119, 9.922946070683342917394260083375, 10.82064368787251089401295184476

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