Properties

Label 2-684-19.18-c0-0-2
Degree $2$
Conductor $684$
Sign $1$
Analytic cond. $0.341360$
Root an. cond. $0.584260$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·5-s + 7-s − 1.73·11-s − 1.73·17-s − 19-s + 1.99·25-s + 1.73·35-s − 43-s + 1.73·47-s − 2.99·55-s + 61-s − 73-s − 1.73·77-s − 2.99·85-s − 1.73·95-s − 1.73·119-s + ⋯
L(s)  = 1  + 1.73·5-s + 7-s − 1.73·11-s − 1.73·17-s − 19-s + 1.99·25-s + 1.73·35-s − 43-s + 1.73·47-s − 2.99·55-s + 61-s − 73-s − 1.73·77-s − 2.99·85-s − 1.73·95-s − 1.73·119-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(0.341360\)
Root analytic conductor: \(0.584260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (37, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.187721470\)
\(L(\frac12)\) \(\approx\) \(1.187721470\)
\(L(1)\) 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 \( 1 - 1.73T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + 1.73T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.73T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - 1.73T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.63819076265861274386595416207, −9.986579019578883302601213491787, −8.906192865166512475222754760476, −8.292569501938580457560570263668, −7.09915660721554118584910851156, −6.10394053929476773795301998781, −5.27195744699599578733541998541, −4.54795630381886956161819096058, −2.54548297162908671539998415590, −1.94363267413058732322543134869, 1.94363267413058732322543134869, 2.54548297162908671539998415590, 4.54795630381886956161819096058, 5.27195744699599578733541998541, 6.10394053929476773795301998781, 7.09915660721554118584910851156, 8.292569501938580457560570263668, 8.906192865166512475222754760476, 9.986579019578883302601213491787, 10.63819076265861274386595416207

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