Properties

Label 2-3381-1.1-c1-0-124
Degree $2$
Conductor $3381$
Sign $-1$
Analytic cond. $26.9974$
Root an. cond. $5.19590$
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  + 0.488·2-s + 3-s − 1.76·4-s + 0.529·5-s + 0.488·6-s − 1.83·8-s + 9-s + 0.258·10-s − 1.22·11-s − 1.76·12-s + 1.15·13-s + 0.529·15-s + 2.62·16-s − 6.79·17-s + 0.488·18-s + 2.34·19-s − 0.933·20-s − 0.599·22-s + 23-s − 1.83·24-s − 4.71·25-s + 0.561·26-s + 27-s − 5.30·29-s + 0.258·30-s + 10.3·31-s + 4.95·32-s + ⋯
L(s)  = 1  + 0.345·2-s + 0.577·3-s − 0.880·4-s + 0.236·5-s + 0.199·6-s − 0.649·8-s + 0.333·9-s + 0.0817·10-s − 0.370·11-s − 0.508·12-s + 0.319·13-s + 0.136·15-s + 0.656·16-s − 1.64·17-s + 0.115·18-s + 0.537·19-s − 0.208·20-s − 0.127·22-s + 0.208·23-s − 0.374·24-s − 0.943·25-s + 0.110·26-s + 0.192·27-s − 0.984·29-s + 0.0472·30-s + 1.85·31-s + 0.875·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(26.9974\)
Root analytic conductor: \(5.19590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3381,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
23 \( 1 - T \)
good2 \( 1 - 0.488T + 2T^{2} \)
5 \( 1 - 0.529T + 5T^{2} \)
11 \( 1 + 1.22T + 11T^{2} \)
13 \( 1 - 1.15T + 13T^{2} \)
17 \( 1 + 6.79T + 17T^{2} \)
19 \( 1 - 2.34T + 19T^{2} \)
29 \( 1 + 5.30T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 - 1.58T + 37T^{2} \)
41 \( 1 - 6.36T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 9.66T + 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 + 7.49T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 - 7.21T + 89T^{2} \)
97 \( 1 + 8.00T + 97T^{2} \)
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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

−8.241717613842556569505610784536, −7.75728266532011323972205557774, −6.58992837655666860359432279876, −6.00016886754608703449026259762, −4.92524337754754993951713354322, −4.46052415418204641397135530896, −3.51462771717133468585385272633, −2.72956104812659916080441465036, −1.57786710814096676856157246516, 0, 1.57786710814096676856157246516, 2.72956104812659916080441465036, 3.51462771717133468585385272633, 4.46052415418204641397135530896, 4.92524337754754993951713354322, 6.00016886754608703449026259762, 6.58992837655666860359432279876, 7.75728266532011323972205557774, 8.241717613842556569505610784536

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