Properties

Label 2-3381-1.1-c1-0-59
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  + 2.51·2-s − 3-s + 4.33·4-s − 2.41·5-s − 2.51·6-s + 5.87·8-s + 9-s − 6.08·10-s + 1.24·11-s − 4.33·12-s − 2.06·13-s + 2.41·15-s + 6.11·16-s + 3.31·17-s + 2.51·18-s + 3.61·19-s − 10.4·20-s + 3.14·22-s − 23-s − 5.87·24-s + 0.848·25-s − 5.20·26-s − 27-s + 4.27·29-s + 6.08·30-s + 7.97·31-s + 3.64·32-s + ⋯
L(s)  = 1  + 1.77·2-s − 0.577·3-s + 2.16·4-s − 1.08·5-s − 1.02·6-s + 2.07·8-s + 0.333·9-s − 1.92·10-s + 0.376·11-s − 1.25·12-s − 0.573·13-s + 0.624·15-s + 1.52·16-s + 0.802·17-s + 0.593·18-s + 0.830·19-s − 2.34·20-s + 0.669·22-s − 0.208·23-s − 1.19·24-s + 0.169·25-s − 1.02·26-s − 0.192·27-s + 0.794·29-s + 1.11·30-s + 1.43·31-s + 0.644·32-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: \(0\)
Selberg data: \((2,\ 3381,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.097025134\)
\(L(\frac12)\) \(\approx\) \(4.097025134\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.51T + 2T^{2} \)
5 \( 1 + 2.41T + 5T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 + 2.06T + 13T^{2} \)
17 \( 1 - 3.31T + 17T^{2} \)
19 \( 1 - 3.61T + 19T^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 - 7.97T + 31T^{2} \)
37 \( 1 - 5.09T + 37T^{2} \)
41 \( 1 - 0.885T + 41T^{2} \)
43 \( 1 - 9.94T + 43T^{2} \)
47 \( 1 - 5.69T + 47T^{2} \)
53 \( 1 + 2.94T + 53T^{2} \)
59 \( 1 + 2.02T + 59T^{2} \)
61 \( 1 - 7.38T + 61T^{2} \)
67 \( 1 - 8.28T + 67T^{2} \)
71 \( 1 + 8.27T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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

−8.231926437194970665443901001414, −7.51433967596099977937615391456, −6.96817959241869282686052403541, −6.09725066243365167913909433919, −5.49994702310512144849380476982, −4.60555035069687604528927604374, −4.17757838461921932182823408996, −3.32649474693494475558368403713, −2.51254438832606926345658424786, −0.977747314436821742847519639486, 0.977747314436821742847519639486, 2.51254438832606926345658424786, 3.32649474693494475558368403713, 4.17757838461921932182823408996, 4.60555035069687604528927604374, 5.49994702310512144849380476982, 6.09725066243365167913909433919, 6.96817959241869282686052403541, 7.51433967596099977937615391456, 8.231926437194970665443901001414

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