Properties

Label 2-1617-1.1-c3-0-121
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.15·2-s − 3·3-s + 18.5·4-s + 4.10·5-s + 15.4·6-s − 54.5·8-s + 9·9-s − 21.1·10-s + 11·11-s − 55.7·12-s + 20.8·13-s − 12.3·15-s + 132.·16-s − 44.4·17-s − 46.3·18-s − 35.8·19-s + 76.1·20-s − 56.7·22-s + 176.·23-s + 163.·24-s − 108.·25-s − 107.·26-s − 27·27-s + 111.·29-s + 63.4·30-s + 2.06·31-s − 246.·32-s + ⋯
L(s)  = 1  − 1.82·2-s − 0.577·3-s + 2.32·4-s + 0.366·5-s + 1.05·6-s − 2.40·8-s + 0.333·9-s − 0.668·10-s + 0.301·11-s − 1.34·12-s + 0.444·13-s − 0.211·15-s + 2.06·16-s − 0.634·17-s − 0.607·18-s − 0.433·19-s + 0.851·20-s − 0.549·22-s + 1.60·23-s + 1.39·24-s − 0.865·25-s − 0.809·26-s − 0.192·27-s + 0.712·29-s + 0.385·30-s + 0.0119·31-s − 1.36·32-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 \( 1 + 5.15T + 8T^{2} \)
5 \( 1 - 4.10T + 125T^{2} \)
13 \( 1 - 20.8T + 2.19e3T^{2} \)
17 \( 1 + 44.4T + 4.91e3T^{2} \)
19 \( 1 + 35.8T + 6.85e3T^{2} \)
23 \( 1 - 176.T + 1.21e4T^{2} \)
29 \( 1 - 111.T + 2.43e4T^{2} \)
31 \( 1 - 2.06T + 2.97e4T^{2} \)
37 \( 1 + 121.T + 5.06e4T^{2} \)
41 \( 1 - 234.T + 6.89e4T^{2} \)
43 \( 1 + 69.4T + 7.95e4T^{2} \)
47 \( 1 + 401.T + 1.03e5T^{2} \)
53 \( 1 + 199.T + 1.48e5T^{2} \)
59 \( 1 + 212.T + 2.05e5T^{2} \)
61 \( 1 - 151.T + 2.26e5T^{2} \)
67 \( 1 + 808.T + 3.00e5T^{2} \)
71 \( 1 - 453.T + 3.57e5T^{2} \)
73 \( 1 + 522.T + 3.89e5T^{2} \)
79 \( 1 + 93.3T + 4.93e5T^{2} \)
83 \( 1 - 666.T + 5.71e5T^{2} \)
89 \( 1 + 1.28e3T + 7.04e5T^{2} \)
97 \( 1 + 844.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.771005284318491725922727419153, −8.033271978173767698826447078904, −7.06315335426560780989785818194, −6.54112969647296031472639871276, −5.76501019413131518272909172297, −4.53723966440087397103651535910, −3.07167225835494115446010642476, −1.92356527848082573519882193980, −1.08132109300361081851997256653, 0, 1.08132109300361081851997256653, 1.92356527848082573519882193980, 3.07167225835494115446010642476, 4.53723966440087397103651535910, 5.76501019413131518272909172297, 6.54112969647296031472639871276, 7.06315335426560780989785818194, 8.033271978173767698826447078904, 8.771005284318491725922727419153

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