Properties

Label 2-819-91.90-c2-0-58
Degree $2$
Conductor $819$
Sign $1$
Analytic cond. $22.3161$
Root an. cond. $4.72399$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 3·5-s + 7·7-s − 13·13-s + 16·16-s + 25·19-s + 12·20-s + 45·23-s − 16·25-s + 28·28-s + 33·29-s − 55·31-s + 21·35-s − 30·41-s − 5·43-s − 81·47-s + 49·49-s − 52·52-s − 15·53-s + 90·59-s + 64·64-s − 39·65-s + 29·73-s + 100·76-s + 67·79-s + 48·80-s + 159·83-s + ⋯
L(s)  = 1  + 4-s + 3/5·5-s + 7-s − 13-s + 16-s + 1.31·19-s + 3/5·20-s + 1.95·23-s − 0.639·25-s + 28-s + 1.13·29-s − 1.77·31-s + 3/5·35-s − 0.731·41-s − 0.116·43-s − 1.72·47-s + 49-s − 52-s − 0.283·53-s + 1.52·59-s + 64-s − 3/5·65-s + 0.397·73-s + 1.31·76-s + 0.848·79-s + 3/5·80-s + 1.91·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(22.3161\)
Root analytic conductor: \(4.72399\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{819} (181, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.073834177\)
\(L(\frac12)\) \(\approx\) \(3.073834177\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - p T \)
13 \( 1 + p T \)
good2 \( ( 1 - p T )( 1 + p T ) \)
5 \( 1 - 3 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 - 25 T + p^{2} T^{2} \)
23 \( 1 - 45 T + p^{2} T^{2} \)
29 \( 1 - 33 T + p^{2} T^{2} \)
31 \( 1 + 55 T + p^{2} T^{2} \)
37 \( ( 1 - p T )( 1 + p T ) \)
41 \( 1 + 30 T + p^{2} T^{2} \)
43 \( 1 + 5 T + p^{2} T^{2} \)
47 \( 1 + 81 T + p^{2} T^{2} \)
53 \( 1 + 15 T + p^{2} T^{2} \)
59 \( 1 - 90 T + p^{2} T^{2} \)
61 \( ( 1 - p T )( 1 + p T ) \)
67 \( ( 1 - p T )( 1 + p T ) \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 29 T + p^{2} T^{2} \)
79 \( 1 - 67 T + p^{2} T^{2} \)
83 \( 1 - 159 T + p^{2} T^{2} \)
89 \( 1 + 165 T + p^{2} T^{2} \)
97 \( 1 + 131 T + p^{2} T^{2} \)
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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

−10.07479443317493452079687038249, −9.327034255135507179103129718718, −8.189730772753198109597348663367, −7.35411260770961415297681626171, −6.74474863434218032927981903734, −5.45848331996311734276499980844, −4.98518240285083309561396704846, −3.32335232912160953566643698905, −2.27380262882479856976688217473, −1.27925895343918578529097919533, 1.27925895343918578529097919533, 2.27380262882479856976688217473, 3.32335232912160953566643698905, 4.98518240285083309561396704846, 5.45848331996311734276499980844, 6.74474863434218032927981903734, 7.35411260770961415297681626171, 8.189730772753198109597348663367, 9.327034255135507179103129718718, 10.07479443317493452079687038249

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