Properties

Label 2-684-1.1-c1-0-6
Degree $2$
Conductor $684$
Sign $-1$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
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  + 5-s − 3·7-s − 5·11-s − 4·13-s + 3·17-s − 19-s − 8·23-s − 4·25-s + 2·29-s + 4·31-s − 3·35-s + 10·37-s − 10·41-s + 43-s + 47-s + 2·49-s + 4·53-s − 5·55-s − 6·59-s − 13·61-s − 4·65-s − 12·67-s − 2·71-s + 9·73-s + 15·77-s + 8·79-s + 12·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.13·7-s − 1.50·11-s − 1.10·13-s + 0.727·17-s − 0.229·19-s − 1.66·23-s − 4/5·25-s + 0.371·29-s + 0.718·31-s − 0.507·35-s + 1.64·37-s − 1.56·41-s + 0.152·43-s + 0.145·47-s + 2/7·49-s + 0.549·53-s − 0.674·55-s − 0.781·59-s − 1.66·61-s − 0.496·65-s − 1.46·67-s − 0.237·71-s + 1.05·73-s + 1.70·77-s + 0.900·79-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{684} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + T \)
good5 \( 1 - T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−9.954901377829594152119131032207, −9.559628331745101664415344236205, −8.146906664648363880391573042940, −7.53862789028003683931489821585, −6.33560598867710411281522365718, −5.65147980657085516848049895381, −4.55834684131140031321784081525, −3.16174551478824360442184908445, −2.25610582463945759407900622794, 0, 2.25610582463945759407900622794, 3.16174551478824360442184908445, 4.55834684131140031321784081525, 5.65147980657085516848049895381, 6.33560598867710411281522365718, 7.53862789028003683931489821585, 8.146906664648363880391573042940, 9.559628331745101664415344236205, 9.954901377829594152119131032207

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