Properties

Label 2-1617-1.1-c3-0-182
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  + 5.29·2-s − 3·3-s + 20.0·4-s − 17.3·5-s − 15.8·6-s + 64.0·8-s + 9·9-s − 92.1·10-s + 11·11-s − 60.2·12-s − 47.9·13-s + 52.1·15-s + 178.·16-s + 83.3·17-s + 47.6·18-s + 81.3·19-s − 349.·20-s + 58.2·22-s − 206.·23-s − 192.·24-s + 177.·25-s − 253.·26-s − 27·27-s + 156.·29-s + 276.·30-s − 246.·31-s + 434.·32-s + ⋯
L(s)  = 1  + 1.87·2-s − 0.577·3-s + 2.51·4-s − 1.55·5-s − 1.08·6-s + 2.82·8-s + 0.333·9-s − 2.91·10-s + 0.301·11-s − 1.44·12-s − 1.02·13-s + 0.898·15-s + 2.79·16-s + 1.18·17-s + 0.624·18-s + 0.982·19-s − 3.90·20-s + 0.564·22-s − 1.86·23-s − 1.63·24-s + 1.41·25-s − 1.91·26-s − 0.192·27-s + 1.00·29-s + 1.68·30-s − 1.43·31-s + 2.39·32-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 \( 1 - 5.29T + 8T^{2} \)
5 \( 1 + 17.3T + 125T^{2} \)
13 \( 1 + 47.9T + 2.19e3T^{2} \)
17 \( 1 - 83.3T + 4.91e3T^{2} \)
19 \( 1 - 81.3T + 6.85e3T^{2} \)
23 \( 1 + 206.T + 1.21e4T^{2} \)
29 \( 1 - 156.T + 2.43e4T^{2} \)
31 \( 1 + 246.T + 2.97e4T^{2} \)
37 \( 1 + 118.T + 5.06e4T^{2} \)
41 \( 1 - 152.T + 6.89e4T^{2} \)
43 \( 1 + 414.T + 7.95e4T^{2} \)
47 \( 1 + 488.T + 1.03e5T^{2} \)
53 \( 1 + 376.T + 1.48e5T^{2} \)
59 \( 1 + 153.T + 2.05e5T^{2} \)
61 \( 1 - 67.8T + 2.26e5T^{2} \)
67 \( 1 + 623.T + 3.00e5T^{2} \)
71 \( 1 + 667.T + 3.57e5T^{2} \)
73 \( 1 - 682.T + 3.89e5T^{2} \)
79 \( 1 + 334.T + 4.93e5T^{2} \)
83 \( 1 + 833.T + 5.71e5T^{2} \)
89 \( 1 + 307.T + 7.04e5T^{2} \)
97 \( 1 - 1.42e3T + 9.12e5T^{2} \)
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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

−8.086420026999378984420219695774, −7.53198697273492402566546690091, −6.89440160114480077734773958974, −5.94171348928996350163490556797, −5.13095803656440453606890099218, −4.48068331736188474807599854544, −3.66961852839462222050034257113, −3.07129349379708694150712926278, −1.60508235462224190135370976272, 0, 1.60508235462224190135370976272, 3.07129349379708694150712926278, 3.66961852839462222050034257113, 4.48068331736188474807599854544, 5.13095803656440453606890099218, 5.94171348928996350163490556797, 6.89440160114480077734773958974, 7.53198697273492402566546690091, 8.086420026999378984420219695774

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