Properties

Label 2-798-1.1-c3-0-49
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 + 5.41·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 10.8·10-s + 17.0·11-s + 12·12-s − 49.6·13-s − 14·14-s + 16.2·15-s + 16·16-s − 110.·17-s − 18·18-s + 19·19-s + 21.6·20-s + 21·21-s − 34.1·22-s − 30.9·23-s − 24·24-s − 95.6·25-s + 99.3·26-s + 27·27-s + 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.484·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.342·10-s + 0.467·11-s + 0.288·12-s − 1.05·13-s − 0.267·14-s + 0.279·15-s + 0.250·16-s − 1.58·17-s − 0.235·18-s + 0.229·19-s + 0.242·20-s + 0.218·21-s − 0.330·22-s − 0.280·23-s − 0.204·24-s − 0.765·25-s + 0.749·26-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 \( 1 - 5.41T + 125T^{2} \)
11 \( 1 - 17.0T + 1.33e3T^{2} \)
13 \( 1 + 49.6T + 2.19e3T^{2} \)
17 \( 1 + 110.T + 4.91e3T^{2} \)
23 \( 1 + 30.9T + 1.21e4T^{2} \)
29 \( 1 + 172.T + 2.43e4T^{2} \)
31 \( 1 + 267.T + 2.97e4T^{2} \)
37 \( 1 - 299.T + 5.06e4T^{2} \)
41 \( 1 - 108.T + 6.89e4T^{2} \)
43 \( 1 + 322.T + 7.95e4T^{2} \)
47 \( 1 + 426.T + 1.03e5T^{2} \)
53 \( 1 + 136.T + 1.48e5T^{2} \)
59 \( 1 - 303.T + 2.05e5T^{2} \)
61 \( 1 - 419.T + 2.26e5T^{2} \)
67 \( 1 + 709.T + 3.00e5T^{2} \)
71 \( 1 - 37.6T + 3.57e5T^{2} \)
73 \( 1 - 836.T + 3.89e5T^{2} \)
79 \( 1 + 52.3T + 4.93e5T^{2} \)
83 \( 1 + 762.T + 5.71e5T^{2} \)
89 \( 1 + 406.T + 7.04e5T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−9.442339699112908368546813392859, −8.756449709106958247689353917212, −7.79037787494325622105102366005, −7.08989259955434359846419116113, −6.12162998910597024856959408668, −4.94561268003871861868810050121, −3.80026894394417091995783385595, −2.39215471760742314962668481036, −1.71767966597873652647834231975, 0, 1.71767966597873652647834231975, 2.39215471760742314962668481036, 3.80026894394417091995783385595, 4.94561268003871861868810050121, 6.12162998910597024856959408668, 7.08989259955434359846419116113, 7.79037787494325622105102366005, 8.756449709106958247689353917212, 9.442339699112908368546813392859

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