Properties

Label 2-3381-1.1-c1-0-79
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.79·2-s − 3-s + 5.83·4-s − 1.94·5-s − 2.79·6-s + 10.7·8-s + 9-s − 5.45·10-s − 2.68·11-s − 5.83·12-s + 4.19·13-s + 1.94·15-s + 18.4·16-s − 0.632·17-s + 2.79·18-s + 5.55·19-s − 11.3·20-s − 7.52·22-s + 23-s − 10.7·24-s − 1.20·25-s + 11.7·26-s − 27-s + 5.57·29-s + 5.45·30-s − 5.59·31-s + 30.0·32-s + ⋯
L(s)  = 1  + 1.97·2-s − 0.577·3-s + 2.91·4-s − 0.870·5-s − 1.14·6-s + 3.79·8-s + 0.333·9-s − 1.72·10-s − 0.810·11-s − 1.68·12-s + 1.16·13-s + 0.502·15-s + 4.60·16-s − 0.153·17-s + 0.659·18-s + 1.27·19-s − 2.54·20-s − 1.60·22-s + 0.208·23-s − 2.19·24-s − 0.241·25-s + 2.30·26-s − 0.192·27-s + 1.03·29-s + 0.995·30-s − 1.00·31-s + 5.30·32-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: $\chi_{3381} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.550198657\)
\(L(\frac12)\) \(\approx\) \(5.550198657\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.79T + 2T^{2} \)
5 \( 1 + 1.94T + 5T^{2} \)
11 \( 1 + 2.68T + 11T^{2} \)
13 \( 1 - 4.19T + 13T^{2} \)
17 \( 1 + 0.632T + 17T^{2} \)
19 \( 1 - 5.55T + 19T^{2} \)
29 \( 1 - 5.57T + 29T^{2} \)
31 \( 1 + 5.59T + 31T^{2} \)
37 \( 1 - 4.11T + 37T^{2} \)
41 \( 1 - 5.19T + 41T^{2} \)
43 \( 1 + 1.98T + 43T^{2} \)
47 \( 1 + 0.292T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 - 7.31T + 59T^{2} \)
61 \( 1 + 6.28T + 61T^{2} \)
67 \( 1 - 3.14T + 67T^{2} \)
71 \( 1 - 6.00T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 - 1.81T + 89T^{2} \)
97 \( 1 + 2.95T + 97T^{2} \)
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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

−7.996979116748602361505512695981, −7.72803562955776671281632738360, −6.76714568687414919770589726647, −6.19418067624137045630841637865, −5.34973526148071827525724271902, −4.88915681598887839469927738487, −3.92163415094295161143472542549, −3.45097530018005864287317100911, −2.46706805048933552252581574356, −1.15468592851428619226280594731, 1.15468592851428619226280594731, 2.46706805048933552252581574356, 3.45097530018005864287317100911, 3.92163415094295161143472542549, 4.88915681598887839469927738487, 5.34973526148071827525724271902, 6.19418067624137045630841637865, 6.76714568687414919770589726647, 7.72803562955776671281632738360, 7.996979116748602361505512695981

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