Properties

Label 2-10304-1.1-c1-0-37
Degree $2$
Conductor $10304$
Sign $1$
Analytic cond. $82.2778$
Root an. cond. $9.07071$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s − 3·9-s − 4·11-s − 6·13-s − 2·17-s − 4·19-s − 23-s − 25-s + 2·29-s − 4·31-s − 2·35-s + 2·37-s − 6·41-s − 12·43-s + 6·45-s − 12·47-s + 49-s + 10·53-s + 8·55-s − 2·61-s − 3·63-s + 12·65-s − 12·67-s + 8·71-s − 14·73-s − 4·77-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 9-s − 1.20·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.208·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.338·35-s + 0.328·37-s − 0.937·41-s − 1.82·43-s + 0.894·45-s − 1.75·47-s + 1/7·49-s + 1.37·53-s + 1.07·55-s − 0.256·61-s − 0.377·63-s + 1.48·65-s − 1.46·67-s + 0.949·71-s − 1.63·73-s − 0.455·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10304\)    =    \(2^{6} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(82.2778\)
Root analytic conductor: \(9.07071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ 1)\)

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 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.06897601824485, −16.61745049352022, −16.08985710630229, −15.21929221913225, −14.91791760325656, −14.62106047492401, −13.59789770418416, −13.24413167516717, −12.39849453880752, −11.91593414656576, −11.48033229571050, −10.78500536278720, −10.24550415853324, −9.584020273509253, −8.661249670246166, −8.219601166433963, −7.741722845245703, −7.082623401979499, −6.345763358441839, −5.350702832493662, −4.968752978501184, −4.265855824838630, −3.308832015751665, −2.601819929423879, −1.911710046298525, 0, 0, 1.911710046298525, 2.601819929423879, 3.308832015751665, 4.265855824838630, 4.968752978501184, 5.350702832493662, 6.345763358441839, 7.082623401979499, 7.741722845245703, 8.219601166433963, 8.661249670246166, 9.584020273509253, 10.24550415853324, 10.78500536278720, 11.48033229571050, 11.91593414656576, 12.39849453880752, 13.24413167516717, 13.59789770418416, 14.62106047492401, 14.91791760325656, 15.21929221913225, 16.08985710630229, 16.61745049352022, 17.06897601824485

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