Properties

Label 2-1617-1.1-c3-0-61
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  − 2.57·2-s − 3·3-s − 1.37·4-s − 19.4·5-s + 7.72·6-s + 24.1·8-s + 9·9-s + 50.0·10-s + 11·11-s + 4.11·12-s − 44.6·13-s + 58.2·15-s − 51.1·16-s − 118.·17-s − 23.1·18-s − 50.3·19-s + 26.6·20-s − 28.3·22-s + 4.82·23-s − 72.3·24-s + 252.·25-s + 115.·26-s − 27·27-s + 62.5·29-s − 150.·30-s − 117.·31-s − 61.3·32-s + ⋯
L(s)  = 1  − 0.910·2-s − 0.577·3-s − 0.171·4-s − 1.73·5-s + 0.525·6-s + 1.06·8-s + 0.333·9-s + 1.58·10-s + 0.301·11-s + 0.0990·12-s − 0.953·13-s + 1.00·15-s − 0.799·16-s − 1.68·17-s − 0.303·18-s − 0.607·19-s + 0.297·20-s − 0.274·22-s + 0.0437·23-s − 0.615·24-s + 2.01·25-s + 0.867·26-s − 0.192·27-s + 0.400·29-s − 0.913·30-s − 0.678·31-s − 0.338·32-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 \( 1 + 2.57T + 8T^{2} \)
5 \( 1 + 19.4T + 125T^{2} \)
13 \( 1 + 44.6T + 2.19e3T^{2} \)
17 \( 1 + 118.T + 4.91e3T^{2} \)
19 \( 1 + 50.3T + 6.85e3T^{2} \)
23 \( 1 - 4.82T + 1.21e4T^{2} \)
29 \( 1 - 62.5T + 2.43e4T^{2} \)
31 \( 1 + 117.T + 2.97e4T^{2} \)
37 \( 1 + 351.T + 5.06e4T^{2} \)
41 \( 1 - 177.T + 6.89e4T^{2} \)
43 \( 1 - 551.T + 7.95e4T^{2} \)
47 \( 1 + 76.4T + 1.03e5T^{2} \)
53 \( 1 - 82.6T + 1.48e5T^{2} \)
59 \( 1 - 508.T + 2.05e5T^{2} \)
61 \( 1 + 341.T + 2.26e5T^{2} \)
67 \( 1 - 367.T + 3.00e5T^{2} \)
71 \( 1 - 169.T + 3.57e5T^{2} \)
73 \( 1 - 784.T + 3.89e5T^{2} \)
79 \( 1 - 284.T + 4.93e5T^{2} \)
83 \( 1 - 672.T + 5.71e5T^{2} \)
89 \( 1 - 72.9T + 7.04e5T^{2} \)
97 \( 1 + 1.14e3T + 9.12e5T^{2} \)
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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

−8.716237584720024332152870185472, −7.85151750633004843709692367800, −7.23835478655269548775805464601, −6.60319979921465565715759561052, −5.08211167503178819087889683783, −4.40158178688799051442583528008, −3.79777598604350944498619087052, −2.20611417467040612878874864875, −0.69417021123350392705640481699, 0, 0.69417021123350392705640481699, 2.20611417467040612878874864875, 3.79777598604350944498619087052, 4.40158178688799051442583528008, 5.08211167503178819087889683783, 6.60319979921465565715759561052, 7.23835478655269548775805464601, 7.85151750633004843709692367800, 8.716237584720024332152870185472

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