Properties

Label 2-798-1.1-c3-0-48
Degree $2$
Conductor $798$
Sign $-1$
Analytic cond. $47.0835$
Root an. cond. $6.86174$
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  − 2·2-s + 3·3-s + 4·4-s + 11.9·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s − 23.8·10-s − 9.57·11-s + 12·12-s − 59.5·13-s + 14·14-s + 35.7·15-s + 16·16-s + 52.8·17-s − 18·18-s − 19·19-s + 47.7·20-s − 21·21-s + 19.1·22-s − 103.·23-s − 24·24-s + 17.3·25-s + 119.·26-s + 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.06·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.754·10-s − 0.262·11-s + 0.288·12-s − 1.26·13-s + 0.267·14-s + 0.616·15-s + 0.250·16-s + 0.753·17-s − 0.235·18-s − 0.229·19-s + 0.533·20-s − 0.218·21-s + 0.185·22-s − 0.939·23-s − 0.204·24-s + 0.138·25-s + 0.897·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $-1$
Analytic conductor: \(47.0835\)
Root analytic conductor: \(6.86174\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 798,\ (\ :3/2),\ -1)\)

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)$
bad2 \( 1 + 2T \)
3 \( 1 - 3T \)
7 \( 1 + 7T \)
19 \( 1 + 19T \)
good5 \( 1 - 11.9T + 125T^{2} \)
11 \( 1 + 9.57T + 1.33e3T^{2} \)
13 \( 1 + 59.5T + 2.19e3T^{2} \)
17 \( 1 - 52.8T + 4.91e3T^{2} \)
23 \( 1 + 103.T + 1.21e4T^{2} \)
29 \( 1 + 182.T + 2.43e4T^{2} \)
31 \( 1 + 99.0T + 2.97e4T^{2} \)
37 \( 1 + 301.T + 5.06e4T^{2} \)
41 \( 1 + 100.T + 6.89e4T^{2} \)
43 \( 1 + 140.T + 7.95e4T^{2} \)
47 \( 1 - 234.T + 1.03e5T^{2} \)
53 \( 1 - 253.T + 1.48e5T^{2} \)
59 \( 1 + 410.T + 2.05e5T^{2} \)
61 \( 1 + 250.T + 2.26e5T^{2} \)
67 \( 1 - 272.T + 3.00e5T^{2} \)
71 \( 1 + 332.T + 3.57e5T^{2} \)
73 \( 1 - 544.T + 3.89e5T^{2} \)
79 \( 1 - 879.T + 4.93e5T^{2} \)
83 \( 1 + 200.T + 5.71e5T^{2} \)
89 \( 1 + 954.T + 7.04e5T^{2} \)
97 \( 1 + 1.39e3T + 9.12e5T^{2} \)
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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

−9.612029698837025473311530061349, −8.773158267228061085172086766732, −7.75737096618784202256028070463, −7.07965234174733616014271917819, −6.00059661493869955474218710313, −5.16707838113381127767239126593, −3.63110788759666411519177057018, −2.46376941903874804906698979457, −1.71521295673885339944884725488, 0, 1.71521295673885339944884725488, 2.46376941903874804906698979457, 3.63110788759666411519177057018, 5.16707838113381127767239126593, 6.00059661493869955474218710313, 7.07965234174733616014271917819, 7.75737096618784202256028070463, 8.773158267228061085172086766732, 9.612029698837025473311530061349

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