Properties

Label 2-693-1.1-c1-0-18
Degree $2$
Conductor $693$
Sign $-1$
Analytic cond. $5.53363$
Root an. cond. $2.35236$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 5-s − 7-s + 11-s − 4·13-s + 4·16-s − 2·17-s − 6·19-s − 2·20-s + 5·23-s − 4·25-s + 2·28-s − 10·29-s + 31-s − 35-s − 5·37-s + 2·41-s − 8·43-s − 2·44-s − 8·47-s + 49-s + 8·52-s + 6·53-s + 55-s − 3·59-s − 2·61-s − 8·64-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 0.377·7-s + 0.301·11-s − 1.10·13-s + 16-s − 0.485·17-s − 1.37·19-s − 0.447·20-s + 1.04·23-s − 4/5·25-s + 0.377·28-s − 1.85·29-s + 0.179·31-s − 0.169·35-s − 0.821·37-s + 0.312·41-s − 1.21·43-s − 0.301·44-s − 1.16·47-s + 1/7·49-s + 1.10·52-s + 0.824·53-s + 0.134·55-s − 0.390·59-s − 0.256·61-s − 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 - T \)
good2 \( 1 + p T^{2} \)
5 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−9.843866131006304498608069072697, −9.263232139707367468574678849511, −8.513651119993219671602342798866, −7.41775134048211629573821110448, −6.42370236549041572726553632076, −5.38913047752314397636287568574, −4.53418395180193611114078817717, −3.48522622489004363314719629746, −2.00735365908684561839702035924, 0, 2.00735365908684561839702035924, 3.48522622489004363314719629746, 4.53418395180193611114078817717, 5.38913047752314397636287568574, 6.42370236549041572726553632076, 7.41775134048211629573821110448, 8.513651119993219671602342798866, 9.263232139707367468574678849511, 9.843866131006304498608069072697

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