Properties

Label 2-1617-1.1-c1-0-19
Degree $2$
Conductor $1617$
Sign $1$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
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  − 2.22·2-s − 3-s + 2.96·4-s + 4.15·5-s + 2.22·6-s − 2.15·8-s + 9-s − 9.25·10-s + 11-s − 2.96·12-s − 2.93·13-s − 4.15·15-s − 1.13·16-s + 1.36·17-s − 2.22·18-s + 4.10·19-s + 12.3·20-s − 2.22·22-s + 2.93·23-s + 2.15·24-s + 12.2·25-s + 6.53·26-s − 27-s − 8.97·29-s + 9.25·30-s + 4.10·31-s + 6.83·32-s + ⋯
L(s)  = 1  − 1.57·2-s − 0.577·3-s + 1.48·4-s + 1.85·5-s + 0.909·6-s − 0.760·8-s + 0.333·9-s − 2.92·10-s + 0.301·11-s − 0.856·12-s − 0.812·13-s − 1.07·15-s − 0.284·16-s + 0.331·17-s − 0.525·18-s + 0.941·19-s + 2.75·20-s − 0.475·22-s + 0.612·23-s + 0.439·24-s + 2.44·25-s + 1.28·26-s − 0.192·27-s − 1.66·29-s + 1.68·30-s + 0.737·31-s + 1.20·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9610492374\)
\(L(\frac12)\) \(\approx\) \(0.9610492374\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + 2.22T + 2T^{2} \)
5 \( 1 - 4.15T + 5T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 - 1.36T + 17T^{2} \)
19 \( 1 - 4.10T + 19T^{2} \)
23 \( 1 - 2.93T + 23T^{2} \)
29 \( 1 + 8.97T + 29T^{2} \)
31 \( 1 - 4.10T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + 6.60T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 6.22T + 47T^{2} \)
53 \( 1 + 6.71T + 53T^{2} \)
59 \( 1 + 0.797T + 59T^{2} \)
61 \( 1 + 7.77T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 + 5.68T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + 4.38T + 79T^{2} \)
83 \( 1 - 6.01T + 83T^{2} \)
89 \( 1 - 1.32T + 89T^{2} \)
97 \( 1 - 2.57T + 97T^{2} \)
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

−9.364454846431990368249901306548, −9.117448667808617407982780788616, −7.77268713137031912356768838131, −7.15464995512273697848814936674, −6.25866138484632604593617213660, −5.62362667671519481931827854912, −4.68659465608188143908176599984, −2.80676600450042750474140673821, −1.84786906912170371429727111946, −0.943943384607049759458929734095, 0.943943384607049759458929734095, 1.84786906912170371429727111946, 2.80676600450042750474140673821, 4.68659465608188143908176599984, 5.62362667671519481931827854912, 6.25866138484632604593617213660, 7.15464995512273697848814936674, 7.77268713137031912356768838131, 9.117448667808617407982780788616, 9.364454846431990368249901306548

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