Properties

Label 2-1617-1.1-c3-0-191
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  + 4.13·2-s − 3·3-s + 9.09·4-s + 10.9·5-s − 12.4·6-s + 4.52·8-s + 9·9-s + 45.3·10-s + 11·11-s − 27.2·12-s − 55.7·13-s − 32.8·15-s − 54.0·16-s + 55.2·17-s + 37.2·18-s − 116.·19-s + 99.6·20-s + 45.4·22-s − 74.0·23-s − 13.5·24-s − 4.89·25-s − 230.·26-s − 27·27-s + 189.·29-s − 135.·30-s − 142.·31-s − 259.·32-s + ⋯
L(s)  = 1  + 1.46·2-s − 0.577·3-s + 1.13·4-s + 0.980·5-s − 0.843·6-s + 0.199·8-s + 0.333·9-s + 1.43·10-s + 0.301·11-s − 0.656·12-s − 1.18·13-s − 0.565·15-s − 0.844·16-s + 0.788·17-s + 0.487·18-s − 1.40·19-s + 1.11·20-s + 0.440·22-s − 0.671·23-s − 0.115·24-s − 0.0391·25-s − 1.73·26-s − 0.192·27-s + 1.21·29-s − 0.827·30-s − 0.824·31-s − 1.43·32-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 \( 1 - 4.13T + 8T^{2} \)
5 \( 1 - 10.9T + 125T^{2} \)
13 \( 1 + 55.7T + 2.19e3T^{2} \)
17 \( 1 - 55.2T + 4.91e3T^{2} \)
19 \( 1 + 116.T + 6.85e3T^{2} \)
23 \( 1 + 74.0T + 1.21e4T^{2} \)
29 \( 1 - 189.T + 2.43e4T^{2} \)
31 \( 1 + 142.T + 2.97e4T^{2} \)
37 \( 1 - 241.T + 5.06e4T^{2} \)
41 \( 1 + 215.T + 6.89e4T^{2} \)
43 \( 1 + 252.T + 7.95e4T^{2} \)
47 \( 1 + 557.T + 1.03e5T^{2} \)
53 \( 1 - 170.T + 1.48e5T^{2} \)
59 \( 1 + 791.T + 2.05e5T^{2} \)
61 \( 1 - 257.T + 2.26e5T^{2} \)
67 \( 1 - 219.T + 3.00e5T^{2} \)
71 \( 1 + 12.4T + 3.57e5T^{2} \)
73 \( 1 - 107.T + 3.89e5T^{2} \)
79 \( 1 - 11.9T + 4.93e5T^{2} \)
83 \( 1 - 509.T + 5.71e5T^{2} \)
89 \( 1 + 1.35e3T + 7.04e5T^{2} \)
97 \( 1 - 405.T + 9.12e5T^{2} \)
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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

−8.658935316565142459841769356433, −7.51157032906736987505425869723, −6.41116456728552975051436399266, −6.19551432211699516704364355991, −5.18389330989272950790400077472, −4.70748952707092407487881985895, −3.70728616869470547872296963422, −2.58478408222483094482317039109, −1.72877096203452586449679787632, 0, 1.72877096203452586449679787632, 2.58478408222483094482317039109, 3.70728616869470547872296963422, 4.70748952707092407487881985895, 5.18389330989272950790400077472, 6.19551432211699516704364355991, 6.41116456728552975051436399266, 7.51157032906736987505425869723, 8.658935316565142459841769356433

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