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Properties

Label 2-728-728.181-c0-0-6
Degree $2$
Conductor $728$
Sign $1$
Analytic cond. $0.363319$
Root an. cond. $0.602759$
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  + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s − 11-s + 12-s − 13-s − 14-s + 16-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + 31-s + 32-s − 33-s − 37-s − 39-s + 41-s − 42-s − 44-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s − 11-s + 12-s − 13-s − 14-s + 16-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + 31-s + 32-s − 33-s − 37-s − 39-s + 41-s − 42-s − 44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.363319\)
Root analytic conductor: \(0.602759\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{728} (181, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.954908197\)
\(L(\frac12)\) \(\approx\) \(1.954908197\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 + T \)
good3 \( 1 - T + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 - T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 - T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 + T )^{2} \)
97 \( 1 - T + T^{2} \)
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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

−10.46527913641510387932204173607, −9.921751530672757355761849152303, −8.827908766948804866543626957188, −7.84370019481874175559593267193, −7.13276197202409677581561729415, −6.08354013242749412833078377104, −5.13838333422398699067116549491, −3.98226442988681564534696240726, −2.92419447192281830191792925582, −2.40049192126004654968939139751, 2.40049192126004654968939139751, 2.92419447192281830191792925582, 3.98226442988681564534696240726, 5.13838333422398699067116549491, 6.08354013242749412833078377104, 7.13276197202409677581561729415, 7.84370019481874175559593267193, 8.827908766948804866543626957188, 9.921751530672757355761849152303, 10.46527913641510387932204173607

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