Properties

Label 2-1617-1.1-c1-0-58
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·2-s + 3-s + 4.85·4-s + 2.23·5-s − 2.61·6-s − 7.47·8-s + 9-s − 5.85·10-s − 11-s + 4.85·12-s − 3.47·13-s + 2.23·15-s + 9.85·16-s − 6·17-s − 2.61·18-s + 4.23·19-s + 10.8·20-s + 2.61·22-s − 2.47·23-s − 7.47·24-s + 9.09·26-s + 27-s − 10.2·29-s − 5.85·30-s − 8.47·31-s − 10.8·32-s − 33-s + ⋯
L(s)  = 1  − 1.85·2-s + 0.577·3-s + 2.42·4-s + 0.999·5-s − 1.06·6-s − 2.64·8-s + 0.333·9-s − 1.85·10-s − 0.301·11-s + 1.40·12-s − 0.962·13-s + 0.577·15-s + 2.46·16-s − 1.45·17-s − 0.617·18-s + 0.971·19-s + 2.42·20-s + 0.558·22-s − 0.515·23-s − 1.52·24-s + 1.78·26-s + 0.192·27-s − 1.90·29-s − 1.06·30-s − 1.52·31-s − 1.91·32-s − 0.174·33-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: \(1\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.61T + 2T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
13 \( 1 + 3.47T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 4.23T + 19T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + 8.47T + 31T^{2} \)
37 \( 1 + 0.527T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 0.472T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 6.94T + 61T^{2} \)
67 \( 1 - 0.708T + 67T^{2} \)
71 \( 1 - 4.47T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + 2.47T + 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 - 6.94T + 97T^{2} \)
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

−9.136129125392656469050322013341, −8.410384676094100896686091956837, −7.52275297217827678920123400387, −7.03252398659051344961708471276, −6.08458139083664908339960167012, −5.08833377946787577344317188999, −3.40230264114730994288873085529, −2.16099573002650604749150685811, −1.84069596447255548673900203893, 0, 1.84069596447255548673900203893, 2.16099573002650604749150685811, 3.40230264114730994288873085529, 5.08833377946787577344317188999, 6.08458139083664908339960167012, 7.03252398659051344961708471276, 7.52275297217827678920123400387, 8.410384676094100896686091956837, 9.136129125392656469050322013341

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