Properties

Label 2-1617-1.1-c1-0-2
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  − 0.552·2-s − 3-s − 1.69·4-s − 1.59·5-s + 0.552·6-s + 2.04·8-s + 9-s + 0.878·10-s − 11-s + 1.69·12-s − 2.87·13-s + 1.59·15-s + 2.26·16-s − 4.83·17-s − 0.552·18-s + 1.14·19-s + 2.69·20-s + 0.552·22-s − 3.65·23-s − 2.04·24-s − 2.47·25-s + 1.59·26-s − 27-s − 0.325·29-s − 0.878·30-s − 6.45·31-s − 5.33·32-s + ⋯
L(s)  = 1  − 0.390·2-s − 0.577·3-s − 0.847·4-s − 0.711·5-s + 0.225·6-s + 0.721·8-s + 0.333·9-s + 0.277·10-s − 0.301·11-s + 0.489·12-s − 0.798·13-s + 0.410·15-s + 0.565·16-s − 1.17·17-s − 0.130·18-s + 0.262·19-s + 0.602·20-s + 0.117·22-s − 0.761·23-s − 0.416·24-s − 0.494·25-s + 0.311·26-s − 0.192·27-s − 0.0605·29-s − 0.160·30-s − 1.15·31-s − 0.942·32-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

\(L(1)\) \(\approx\) \(0.4061597978\)
\(L(\frac12)\) \(\approx\) \(0.4061597978\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.552T + 2T^{2} \)
5 \( 1 + 1.59T + 5T^{2} \)
13 \( 1 + 2.87T + 13T^{2} \)
17 \( 1 + 4.83T + 17T^{2} \)
19 \( 1 - 1.14T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 + 0.325T + 29T^{2} \)
31 \( 1 + 6.45T + 31T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 - 4.05T + 41T^{2} \)
43 \( 1 - 4.62T + 43T^{2} \)
47 \( 1 + 0.305T + 47T^{2} \)
53 \( 1 - 5.71T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 + 1.77T + 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 - 9.16T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + 8.71T + 79T^{2} \)
83 \( 1 + 8.40T + 83T^{2} \)
89 \( 1 - 5.74T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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

−9.427569316529826420925907758370, −8.639721717204986415046494635280, −7.74463047582261827146545497942, −7.28613727207526383760163359146, −6.10333610487949633578761473343, −5.16181955077748040061998587935, −4.42540078758503128928314453436, −3.69012740312842942997401141877, −2.11984191104285427172541585450, −0.47318212961588927980077226483, 0.47318212961588927980077226483, 2.11984191104285427172541585450, 3.69012740312842942997401141877, 4.42540078758503128928314453436, 5.16181955077748040061998587935, 6.10333610487949633578761473343, 7.28613727207526383760163359146, 7.74463047582261827146545497942, 8.639721717204986415046494635280, 9.427569316529826420925907758370

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