Properties

Label 2-1617-1.1-c3-0-152
Degree $2$
Conductor $1617$
Sign $-1$
Analytic cond. $95.4060$
Root an. cond. $9.76760$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.48·2-s − 3·3-s + 12.1·4-s + 17.6·5-s + 13.4·6-s − 18.4·8-s + 9·9-s − 79.2·10-s + 11·11-s − 36.3·12-s − 12.6·13-s − 52.9·15-s − 14.0·16-s − 64.0·17-s − 40.3·18-s + 164.·19-s + 214.·20-s − 49.3·22-s − 69.8·23-s + 55.4·24-s + 187.·25-s + 56.8·26-s − 27·27-s + 193.·29-s + 237.·30-s − 23.9·31-s + 210.·32-s + ⋯
L(s)  = 1  − 1.58·2-s − 0.577·3-s + 1.51·4-s + 1.57·5-s + 0.915·6-s − 0.816·8-s + 0.333·9-s − 2.50·10-s + 0.301·11-s − 0.874·12-s − 0.270·13-s − 0.912·15-s − 0.220·16-s − 0.913·17-s − 0.528·18-s + 1.98·19-s + 2.39·20-s − 0.478·22-s − 0.633·23-s + 0.471·24-s + 1.49·25-s + 0.428·26-s − 0.192·27-s + 1.24·29-s + 1.44·30-s − 0.138·31-s + 1.16·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(95.4060\)
Root analytic conductor: \(9.76760\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1617,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
7 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 + 4.48T + 8T^{2} \)
5 \( 1 - 17.6T + 125T^{2} \)
13 \( 1 + 12.6T + 2.19e3T^{2} \)
17 \( 1 + 64.0T + 4.91e3T^{2} \)
19 \( 1 - 164.T + 6.85e3T^{2} \)
23 \( 1 + 69.8T + 1.21e4T^{2} \)
29 \( 1 - 193.T + 2.43e4T^{2} \)
31 \( 1 + 23.9T + 2.97e4T^{2} \)
37 \( 1 + 135.T + 5.06e4T^{2} \)
41 \( 1 + 400.T + 6.89e4T^{2} \)
43 \( 1 + 557.T + 7.95e4T^{2} \)
47 \( 1 + 385.T + 1.03e5T^{2} \)
53 \( 1 + 401.T + 1.48e5T^{2} \)
59 \( 1 - 518.T + 2.05e5T^{2} \)
61 \( 1 + 172.T + 2.26e5T^{2} \)
67 \( 1 + 30.6T + 3.00e5T^{2} \)
71 \( 1 + 300.T + 3.57e5T^{2} \)
73 \( 1 - 90.7T + 3.89e5T^{2} \)
79 \( 1 + 500.T + 4.93e5T^{2} \)
83 \( 1 + 935.T + 5.71e5T^{2} \)
89 \( 1 + 143.T + 7.04e5T^{2} \)
97 \( 1 + 217.T + 9.12e5T^{2} \)
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

−8.816803355931156411665507574977, −8.051452459924352920638793167033, −6.86400986151551844565229703638, −6.61364209221654210968886754497, −5.53076093799294711236169576391, −4.78929903200287141522705762541, −3.03332168931995227294206682650, −1.85409944730917236133064683467, −1.28957328850290534053642282681, 0, 1.28957328850290534053642282681, 1.85409944730917236133064683467, 3.03332168931995227294206682650, 4.78929903200287141522705762541, 5.53076093799294711236169576391, 6.61364209221654210968886754497, 6.86400986151551844565229703638, 8.051452459924352920638793167033, 8.816803355931156411665507574977

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