Properties

Label 2-798-1.1-c3-0-42
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 − 11.4·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 22.9·10-s + 58.8·11-s − 12·12-s + 5.39·13-s − 14·14-s + 34.4·15-s + 16·16-s − 105.·17-s + 18·18-s − 19·19-s − 45.9·20-s + 21·21-s + 117.·22-s + 206.·23-s − 24·24-s + 6.70·25-s + 10.7·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.02·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.725·10-s + 1.61·11-s − 0.288·12-s + 0.115·13-s − 0.267·14-s + 0.592·15-s + 0.250·16-s − 1.51·17-s + 0.235·18-s − 0.229·19-s − 0.513·20-s + 0.218·21-s + 1.14·22-s + 1.87·23-s − 0.204·24-s + 0.0536·25-s + 0.0814·26-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 \( 1 + 11.4T + 125T^{2} \)
11 \( 1 - 58.8T + 1.33e3T^{2} \)
13 \( 1 - 5.39T + 2.19e3T^{2} \)
17 \( 1 + 105.T + 4.91e3T^{2} \)
23 \( 1 - 206.T + 1.21e4T^{2} \)
29 \( 1 + 201.T + 2.43e4T^{2} \)
31 \( 1 - 156.T + 2.97e4T^{2} \)
37 \( 1 + 325.T + 5.06e4T^{2} \)
41 \( 1 + 145.T + 6.89e4T^{2} \)
43 \( 1 - 507.T + 7.95e4T^{2} \)
47 \( 1 + 311.T + 1.03e5T^{2} \)
53 \( 1 + 408.T + 1.48e5T^{2} \)
59 \( 1 + 350.T + 2.05e5T^{2} \)
61 \( 1 + 500.T + 2.26e5T^{2} \)
67 \( 1 + 527.T + 3.00e5T^{2} \)
71 \( 1 + 342.T + 3.57e5T^{2} \)
73 \( 1 + 286.T + 3.89e5T^{2} \)
79 \( 1 + 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 672.T + 5.71e5T^{2} \)
89 \( 1 - 407.T + 7.04e5T^{2} \)
97 \( 1 + 844.T + 9.12e5T^{2} \)
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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

−9.356436898872963060835960361960, −8.694955104976785283579712258750, −7.33416666108680395492177942562, −6.77464350387983514999091537537, −6.00433645222493515803387875879, −4.69444779252259583024100655696, −4.09658707766537415518688149846, −3.15066277590454660756381357952, −1.48711479671144251441890311651, 0, 1.48711479671144251441890311651, 3.15066277590454660756381357952, 4.09658707766537415518688149846, 4.69444779252259583024100655696, 6.00433645222493515803387875879, 6.77464350387983514999091537537, 7.33416666108680395492177942562, 8.694955104976785283579712258750, 9.356436898872963060835960361960

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