Properties

Label 2-1617-1.1-c3-0-189
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  + 5.23·2-s − 3·3-s + 19.3·4-s − 10.0·5-s − 15.7·6-s + 59.6·8-s + 9·9-s − 52.5·10-s + 11·11-s − 58.1·12-s − 14.5·13-s + 30.1·15-s + 156.·16-s − 83.6·17-s + 47.1·18-s − 31.1·19-s − 194.·20-s + 57.5·22-s + 64.0·23-s − 178.·24-s − 24.0·25-s − 76.0·26-s − 27·27-s − 245.·29-s + 157.·30-s + 126.·31-s + 344.·32-s + ⋯
L(s)  = 1  + 1.85·2-s − 0.577·3-s + 2.42·4-s − 0.898·5-s − 1.06·6-s + 2.63·8-s + 0.333·9-s − 1.66·10-s + 0.301·11-s − 1.39·12-s − 0.309·13-s + 0.518·15-s + 2.45·16-s − 1.19·17-s + 0.616·18-s − 0.375·19-s − 2.17·20-s + 0.557·22-s + 0.580·23-s − 1.52·24-s − 0.192·25-s − 0.573·26-s − 0.192·27-s − 1.57·29-s + 0.960·30-s + 0.732·31-s + 1.90·32-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 \( 1 - 5.23T + 8T^{2} \)
5 \( 1 + 10.0T + 125T^{2} \)
13 \( 1 + 14.5T + 2.19e3T^{2} \)
17 \( 1 + 83.6T + 4.91e3T^{2} \)
19 \( 1 + 31.1T + 6.85e3T^{2} \)
23 \( 1 - 64.0T + 1.21e4T^{2} \)
29 \( 1 + 245.T + 2.43e4T^{2} \)
31 \( 1 - 126.T + 2.97e4T^{2} \)
37 \( 1 + 237.T + 5.06e4T^{2} \)
41 \( 1 - 325.T + 6.89e4T^{2} \)
43 \( 1 + 164.T + 7.95e4T^{2} \)
47 \( 1 + 74.3T + 1.03e5T^{2} \)
53 \( 1 + 255.T + 1.48e5T^{2} \)
59 \( 1 + 50.3T + 2.05e5T^{2} \)
61 \( 1 + 68.5T + 2.26e5T^{2} \)
67 \( 1 - 633.T + 3.00e5T^{2} \)
71 \( 1 - 451.T + 3.57e5T^{2} \)
73 \( 1 + 1.13e3T + 3.89e5T^{2} \)
79 \( 1 + 814.T + 4.93e5T^{2} \)
83 \( 1 + 265.T + 5.71e5T^{2} \)
89 \( 1 + 1.14e3T + 7.04e5T^{2} \)
97 \( 1 + 1.08e3T + 9.12e5T^{2} \)
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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

−8.375418947458757660676353872488, −7.32210766848219564107125853046, −6.84440101247913539823537192711, −6.00440297177475070196743051151, −5.18994533554256611055785168806, −4.35862245503092705398787747850, −3.89664777544500573692160737272, −2.82936901491007690229895599255, −1.70977734447768732272070052907, 0, 1.70977734447768732272070052907, 2.82936901491007690229895599255, 3.89664777544500573692160737272, 4.35862245503092705398787747850, 5.18994533554256611055785168806, 6.00440297177475070196743051151, 6.84440101247913539823537192711, 7.32210766848219564107125853046, 8.375418947458757660676353872488

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