Properties

Label 2-1617-1.1-c3-0-142
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  − 1.36·2-s − 3·3-s − 6.14·4-s + 14.9·5-s + 4.09·6-s + 19.2·8-s + 9·9-s − 20.3·10-s + 11·11-s + 18.4·12-s − 36.0·13-s − 44.7·15-s + 22.8·16-s + 59.4·17-s − 12.2·18-s − 69.7·19-s − 91.6·20-s − 15.0·22-s − 29.1·23-s − 57.8·24-s + 97.8·25-s + 49.1·26-s − 27·27-s + 34.4·29-s + 61.0·30-s + 309.·31-s − 185.·32-s + ⋯
L(s)  = 1  − 0.482·2-s − 0.577·3-s − 0.767·4-s + 1.33·5-s + 0.278·6-s + 0.852·8-s + 0.333·9-s − 0.643·10-s + 0.301·11-s + 0.443·12-s − 0.768·13-s − 0.770·15-s + 0.356·16-s + 0.848·17-s − 0.160·18-s − 0.841·19-s − 1.02·20-s − 0.145·22-s − 0.264·23-s − 0.492·24-s + 0.782·25-s + 0.370·26-s − 0.192·27-s + 0.220·29-s + 0.371·30-s + 1.79·31-s − 1.02·32-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 \( 1 + 1.36T + 8T^{2} \)
5 \( 1 - 14.9T + 125T^{2} \)
13 \( 1 + 36.0T + 2.19e3T^{2} \)
17 \( 1 - 59.4T + 4.91e3T^{2} \)
19 \( 1 + 69.7T + 6.85e3T^{2} \)
23 \( 1 + 29.1T + 1.21e4T^{2} \)
29 \( 1 - 34.4T + 2.43e4T^{2} \)
31 \( 1 - 309.T + 2.97e4T^{2} \)
37 \( 1 + 347.T + 5.06e4T^{2} \)
41 \( 1 + 407.T + 6.89e4T^{2} \)
43 \( 1 + 168.T + 7.95e4T^{2} \)
47 \( 1 + 568.T + 1.03e5T^{2} \)
53 \( 1 + 146.T + 1.48e5T^{2} \)
59 \( 1 - 458.T + 2.05e5T^{2} \)
61 \( 1 - 638.T + 2.26e5T^{2} \)
67 \( 1 + 179.T + 3.00e5T^{2} \)
71 \( 1 + 31.5T + 3.57e5T^{2} \)
73 \( 1 + 187.T + 3.89e5T^{2} \)
79 \( 1 - 939.T + 4.93e5T^{2} \)
83 \( 1 - 620.T + 5.71e5T^{2} \)
89 \( 1 - 775.T + 7.04e5T^{2} \)
97 \( 1 + 297.T + 9.12e5T^{2} \)
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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

−8.694316728726191429276449320805, −8.061970690671633742474670003992, −6.87842338781886373619408416053, −6.23695972690092341113581877109, −5.20767530487883282304478889045, −4.81663687012568181343121815271, −3.53946757281041759806558362573, −2.10759791365528759944463759770, −1.21107056740382640596734994596, 0, 1.21107056740382640596734994596, 2.10759791365528759944463759770, 3.53946757281041759806558362573, 4.81663687012568181343121815271, 5.20767530487883282304478889045, 6.23695972690092341113581877109, 6.87842338781886373619408416053, 8.061970690671633742474670003992, 8.694316728726191429276449320805

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