Properties

Label 2-3381-1.1-c1-0-20
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.05·2-s − 3-s − 0.890·4-s + 2.17·5-s + 1.05·6-s + 3.04·8-s + 9-s − 2.29·10-s − 2.62·11-s + 0.890·12-s − 3.30·13-s − 2.17·15-s − 1.42·16-s − 2.96·17-s − 1.05·18-s + 0.697·19-s − 1.93·20-s + 2.76·22-s + 23-s − 3.04·24-s − 0.264·25-s + 3.48·26-s − 27-s + 7.31·29-s + 2.29·30-s + 5.60·31-s − 4.58·32-s + ⋯
L(s)  = 1  − 0.744·2-s − 0.577·3-s − 0.445·4-s + 0.973·5-s + 0.430·6-s + 1.07·8-s + 0.333·9-s − 0.724·10-s − 0.790·11-s + 0.256·12-s − 0.917·13-s − 0.561·15-s − 0.356·16-s − 0.720·17-s − 0.248·18-s + 0.160·19-s − 0.433·20-s + 0.589·22-s + 0.208·23-s − 0.621·24-s − 0.0529·25-s + 0.683·26-s − 0.192·27-s + 1.35·29-s + 0.418·30-s + 1.00·31-s − 0.810·32-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

\(L(1)\) \(\approx\) \(0.7750859986\)
\(L(\frac12)\) \(\approx\) \(0.7750859986\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.05T + 2T^{2} \)
5 \( 1 - 2.17T + 5T^{2} \)
11 \( 1 + 2.62T + 11T^{2} \)
13 \( 1 + 3.30T + 13T^{2} \)
17 \( 1 + 2.96T + 17T^{2} \)
19 \( 1 - 0.697T + 19T^{2} \)
29 \( 1 - 7.31T + 29T^{2} \)
31 \( 1 - 5.60T + 31T^{2} \)
37 \( 1 + 4.52T + 37T^{2} \)
41 \( 1 - 2.28T + 41T^{2} \)
43 \( 1 + 7.79T + 43T^{2} \)
47 \( 1 + 3.81T + 47T^{2} \)
53 \( 1 - 3.13T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 - 1.48T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 0.705T + 73T^{2} \)
79 \( 1 + 8.60T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + 0.333T + 89T^{2} \)
97 \( 1 - 0.124T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.569118752759111215275319635165, −8.097781076750739873256111172444, −7.07210998269816753830922291291, −6.53043785892325795312855853424, −5.38346579134180212653267153611, −5.03992850171119329975317930736, −4.17178921218129550888082215515, −2.72832929673530371904236302188, −1.82610538665401551553390315131, −0.60451917206128996640654774511, 0.60451917206128996640654774511, 1.82610538665401551553390315131, 2.72832929673530371904236302188, 4.17178921218129550888082215515, 5.03992850171119329975317930736, 5.38346579134180212653267153611, 6.53043785892325795312855853424, 7.07210998269816753830922291291, 8.097781076750739873256111172444, 8.569118752759111215275319635165

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