Properties

Label 2-798-1.1-c3-0-43
Degree $2$
Conductor $798$
Sign $-1$
Analytic cond. $47.0835$
Root an. cond. $6.86174$
Motivic weight $3$
Arithmetic yes
Rational yes
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 − 10·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 20·10-s + 8·11-s − 12·12-s − 50·13-s + 14·14-s + 30·15-s + 16·16-s + 114·17-s + 18·18-s + 19·19-s − 40·20-s − 21·21-s + 16·22-s − 148·23-s − 24·24-s − 25·25-s − 100·26-s − 27·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.219·11-s − 0.288·12-s − 1.06·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 1.62·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s − 0.218·21-s + 0.155·22-s − 1.34·23-s − 0.204·24-s − 1/5·25-s − 0.754·26-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: yes
Arithmetic: yes
Character: $\chi_{798} (1, \cdot )$
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 \( 1 - p T \)
3 \( 1 + p T \)
7 \( 1 - p T \)
19 \( 1 - p T \)
good5 \( 1 + 2 p T + p^{3} T^{2} \)
11 \( 1 - 8 T + p^{3} T^{2} \)
13 \( 1 + 50 T + p^{3} T^{2} \)
17 \( 1 - 114 T + p^{3} T^{2} \)
23 \( 1 + 148 T + p^{3} T^{2} \)
29 \( 1 + 30 T + p^{3} T^{2} \)
31 \( 1 - 304 T + p^{3} T^{2} \)
37 \( 1 + 274 T + p^{3} T^{2} \)
41 \( 1 + 202 T + p^{3} T^{2} \)
43 \( 1 + 116 T + p^{3} T^{2} \)
47 \( 1 + 324 T + p^{3} T^{2} \)
53 \( 1 + 550 T + p^{3} T^{2} \)
59 \( 1 - 628 T + p^{3} T^{2} \)
61 \( 1 + 58 T + p^{3} T^{2} \)
67 \( 1 + 756 T + p^{3} T^{2} \)
71 \( 1 + 216 T + p^{3} T^{2} \)
73 \( 1 + 278 T + p^{3} T^{2} \)
79 \( 1 + 952 T + p^{3} T^{2} \)
83 \( 1 + 1184 T + p^{3} T^{2} \)
89 \( 1 - 1542 T + p^{3} T^{2} \)
97 \( 1 + 870 T + p^{3} T^{2} \)
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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

−9.828659954233367664716742925187, −8.228738366472281490054291905926, −7.67865611132199113868618495893, −6.79242422931843635759220717821, −5.73333181104854334627083748114, −4.91771851307882802899011710548, −4.07350671509007715158500917271, −3.06467973698718390096325925740, −1.53437772998690697228117098604, 0, 1.53437772998690697228117098604, 3.06467973698718390096325925740, 4.07350671509007715158500917271, 4.91771851307882802899011710548, 5.73333181104854334627083748114, 6.79242422931843635759220717821, 7.67865611132199113868618495893, 8.228738366472281490054291905926, 9.828659954233367664716742925187

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