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Properties

Label 2-684-19.18-c0-0-1
Degree $2$
Conductor $684$
Sign $1$
Analytic cond. $0.341360$
Root an. cond. $0.584260$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s + 11-s + 17-s + 19-s − 2·23-s − 35-s − 43-s + 47-s + 55-s − 61-s − 73-s − 77-s − 2·83-s + 85-s + 95-s − 2·101-s − 2·115-s − 119-s + ⋯
L(s)  = 1  + 5-s − 7-s + 11-s + 17-s + 19-s − 2·23-s − 35-s − 43-s + 47-s + 55-s − 61-s − 73-s − 77-s − 2·83-s + 85-s + 95-s − 2·101-s − 2·115-s − 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: yes
Arithmetic: yes
Character: $\chi_{684} (37, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.049806151\)
\(L(\frac12)\) \(\approx\) \(1.049806151\)
\(L(1)\) 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 \( 1 - T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 - T + T^{2} \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 + T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−10.35491111494555031411626029725, −9.758818438700118784693109349550, −9.314387812062134939433397223502, −8.117394606784884318841001905600, −7.03087957755923141913663523242, −6.10749108764499485087312327875, −5.59033728809514506575945817746, −4.06732770039287226268176886749, −3.05201959040369072688091579245, −1.62495726525899861887525643650, 1.62495726525899861887525643650, 3.05201959040369072688091579245, 4.06732770039287226268176886749, 5.59033728809514506575945817746, 6.10749108764499485087312327875, 7.03087957755923141913663523242, 8.117394606784884318841001905600, 9.314387812062134939433397223502, 9.758818438700118784693109349550, 10.35491111494555031411626029725

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