Properties

Label 2-798-1.1-c3-0-22
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 − 16.2·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 32.5·10-s + 29.7·11-s − 12·12-s − 46.5·13-s + 14·14-s + 48.8·15-s + 16·16-s − 12.5·17-s − 18·18-s + 19·19-s − 65.1·20-s + 21·21-s − 59.5·22-s + 113.·23-s + 24·24-s + 140.·25-s + 93.1·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.45·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.03·10-s + 0.816·11-s − 0.288·12-s − 0.993·13-s + 0.267·14-s + 0.841·15-s + 0.250·16-s − 0.178·17-s − 0.235·18-s + 0.229·19-s − 0.728·20-s + 0.218·21-s − 0.577·22-s + 1.02·23-s + 0.204·24-s + 1.12·25-s + 0.702·26-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 \( 1 + 16.2T + 125T^{2} \)
11 \( 1 - 29.7T + 1.33e3T^{2} \)
13 \( 1 + 46.5T + 2.19e3T^{2} \)
17 \( 1 + 12.5T + 4.91e3T^{2} \)
23 \( 1 - 113.T + 1.21e4T^{2} \)
29 \( 1 - 156.T + 2.43e4T^{2} \)
31 \( 1 + 97.1T + 2.97e4T^{2} \)
37 \( 1 - 230.T + 5.06e4T^{2} \)
41 \( 1 - 204.T + 6.89e4T^{2} \)
43 \( 1 + 41.5T + 7.95e4T^{2} \)
47 \( 1 + 26.4T + 1.03e5T^{2} \)
53 \( 1 - 465.T + 1.48e5T^{2} \)
59 \( 1 + 52.9T + 2.05e5T^{2} \)
61 \( 1 + 32.4T + 2.26e5T^{2} \)
67 \( 1 + 568.T + 3.00e5T^{2} \)
71 \( 1 + 376.T + 3.57e5T^{2} \)
73 \( 1 + 1.03e3T + 3.89e5T^{2} \)
79 \( 1 - 452.T + 4.93e5T^{2} \)
83 \( 1 + 222.T + 5.71e5T^{2} \)
89 \( 1 - 765.T + 7.04e5T^{2} \)
97 \( 1 - 105.T + 9.12e5T^{2} \)
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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

−9.405694502414496120383874223338, −8.655318567592197616504934040307, −7.56287205090650517403840845193, −7.12689816361981885260474888891, −6.17882936044691037959103450225, −4.85283898169297330079317730664, −3.94358173359349410867211565202, −2.78758337654086005500480969957, −1.02446314920538743099175162483, 0, 1.02446314920538743099175162483, 2.78758337654086005500480969957, 3.94358173359349410867211565202, 4.85283898169297330079317730664, 6.17882936044691037959103450225, 7.12689816361981885260474888891, 7.56287205090650517403840845193, 8.655318567592197616504934040307, 9.405694502414496120383874223338

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