Properties

Label 2-1617-1.1-c3-0-181
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.42·2-s + 3·3-s + 21.4·4-s + 16.8·5-s + 16.2·6-s + 72.8·8-s + 9·9-s + 91.3·10-s − 11·11-s + 64.2·12-s − 24.8·13-s + 50.5·15-s + 223.·16-s + 15.9·17-s + 48.8·18-s − 15.1·19-s + 360.·20-s − 59.6·22-s + 17.7·23-s + 218.·24-s + 158.·25-s − 134.·26-s + 27·27-s − 128.·29-s + 274.·30-s − 219.·31-s + 630.·32-s + ⋯
L(s)  = 1  + 1.91·2-s + 0.577·3-s + 2.67·4-s + 1.50·5-s + 1.10·6-s + 3.21·8-s + 0.333·9-s + 2.89·10-s − 0.301·11-s + 1.54·12-s − 0.530·13-s + 0.870·15-s + 3.49·16-s + 0.227·17-s + 0.639·18-s − 0.182·19-s + 4.03·20-s − 0.578·22-s + 0.160·23-s + 1.85·24-s + 1.27·25-s − 1.01·26-s + 0.192·27-s − 0.823·29-s + 1.66·30-s − 1.27·31-s + 3.48·32-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: \(0\)
Selberg data: \((2,\ 1617,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(13.29105286\)
\(L(\frac12)\) \(\approx\) \(13.29105286\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.42T + 8T^{2} \)
5 \( 1 - 16.8T + 125T^{2} \)
13 \( 1 + 24.8T + 2.19e3T^{2} \)
17 \( 1 - 15.9T + 4.91e3T^{2} \)
19 \( 1 + 15.1T + 6.85e3T^{2} \)
23 \( 1 - 17.7T + 1.21e4T^{2} \)
29 \( 1 + 128.T + 2.43e4T^{2} \)
31 \( 1 + 219.T + 2.97e4T^{2} \)
37 \( 1 - 92.0T + 5.06e4T^{2} \)
41 \( 1 - 459.T + 6.89e4T^{2} \)
43 \( 1 - 64.9T + 7.95e4T^{2} \)
47 \( 1 + 497.T + 1.03e5T^{2} \)
53 \( 1 + 526.T + 1.48e5T^{2} \)
59 \( 1 - 578.T + 2.05e5T^{2} \)
61 \( 1 - 221.T + 2.26e5T^{2} \)
67 \( 1 + 860.T + 3.00e5T^{2} \)
71 \( 1 - 580.T + 3.57e5T^{2} \)
73 \( 1 + 510.T + 3.89e5T^{2} \)
79 \( 1 - 1.03e3T + 4.93e5T^{2} \)
83 \( 1 + 606.T + 5.71e5T^{2} \)
89 \( 1 - 23.4T + 7.04e5T^{2} \)
97 \( 1 + 719.T + 9.12e5T^{2} \)
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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

−9.247993038261810193896109912204, −7.889090627620246576375429829966, −7.12510630542045740798318106003, −6.28821728140332338664502453513, −5.60743907355253857025836877960, −4.99417356579414459675528302710, −4.01917259563620011544487403920, −3.00375801437884847587761137736, −2.29391629681456749485592140702, −1.56996547654290206972112084446, 1.56996547654290206972112084446, 2.29391629681456749485592140702, 3.00375801437884847587761137736, 4.01917259563620011544487403920, 4.99417356579414459675528302710, 5.60743907355253857025836877960, 6.28821728140332338664502453513, 7.12510630542045740798318106003, 7.889090627620246576375429829966, 9.247993038261810193896109912204

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