Properties

Label 2-798-1.1-c3-0-23
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s + 4·4-s + 21.5·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 43.0·10-s − 33.5·11-s + 12·12-s − 46.6·13-s − 14·14-s + 64.5·15-s + 16·16-s − 41.8·17-s − 18·18-s − 19·19-s + 86.1·20-s + 21·21-s + 67.0·22-s + 52.5·23-s − 24·24-s + 338.·25-s + 93.3·26-s + 27·27-s + 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.92·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.36·10-s − 0.918·11-s + 0.288·12-s − 0.996·13-s − 0.267·14-s + 1.11·15-s + 0.250·16-s − 0.597·17-s − 0.235·18-s − 0.229·19-s + 0.962·20-s + 0.218·21-s + 0.649·22-s + 0.476·23-s − 0.204·24-s + 2.70·25-s + 0.704·26-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: \(0\)
Selberg data: \((2,\ 798,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.684726476\)
\(L(\frac12)\) \(\approx\) \(2.684726476\)
\(L(\frac{5}{2})\) 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 \( 1 - 21.5T + 125T^{2} \)
11 \( 1 + 33.5T + 1.33e3T^{2} \)
13 \( 1 + 46.6T + 2.19e3T^{2} \)
17 \( 1 + 41.8T + 4.91e3T^{2} \)
23 \( 1 - 52.5T + 1.21e4T^{2} \)
29 \( 1 - 198.T + 2.43e4T^{2} \)
31 \( 1 - 117.T + 2.97e4T^{2} \)
37 \( 1 - 240.T + 5.06e4T^{2} \)
41 \( 1 + 3.25T + 6.89e4T^{2} \)
43 \( 1 - 486.T + 7.95e4T^{2} \)
47 \( 1 + 176.T + 1.03e5T^{2} \)
53 \( 1 - 751.T + 1.48e5T^{2} \)
59 \( 1 - 306.T + 2.05e5T^{2} \)
61 \( 1 - 39.2T + 2.26e5T^{2} \)
67 \( 1 - 582.T + 3.00e5T^{2} \)
71 \( 1 - 586.T + 3.57e5T^{2} \)
73 \( 1 + 759.T + 3.89e5T^{2} \)
79 \( 1 + 300.T + 4.93e5T^{2} \)
83 \( 1 + 826.T + 5.71e5T^{2} \)
89 \( 1 - 455.T + 7.04e5T^{2} \)
97 \( 1 + 528.T + 9.12e5T^{2} \)
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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

−9.962291906364829906743264331424, −9.074229844172003532598177775645, −8.418597743550267119083914220460, −7.36302323682049974800606716779, −6.50887901282638452657954649451, −5.53688647781536747230479761819, −4.65372271397177601728398297227, −2.55247184454947843764340915107, −2.40577868878851121014792129784, −1.02734491058775931517426532333, 1.02734491058775931517426532333, 2.40577868878851121014792129784, 2.55247184454947843764340915107, 4.65372271397177601728398297227, 5.53688647781536747230479761819, 6.50887901282638452657954649451, 7.36302323682049974800606716779, 8.418597743550267119083914220460, 9.074229844172003532598177775645, 9.962291906364829906743264331424

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