Properties

Label 2-10304-1.1-c1-0-27
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s − 2·9-s + 6·11-s + 3·13-s − 21-s + 23-s − 5·25-s + 5·27-s + 3·29-s − 7·31-s − 6·33-s − 8·37-s − 3·39-s − 11·41-s − 4·43-s + 47-s + 49-s − 4·53-s − 12·59-s + 6·61-s − 2·63-s − 12·67-s − 69-s − 5·71-s + 15·73-s + 5·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.80·11-s + 0.832·13-s − 0.218·21-s + 0.208·23-s − 25-s + 0.962·27-s + 0.557·29-s − 1.25·31-s − 1.04·33-s − 1.31·37-s − 0.480·39-s − 1.71·41-s − 0.609·43-s + 0.145·47-s + 1/7·49-s − 0.549·53-s − 1.56·59-s + 0.768·61-s − 0.251·63-s − 1.46·67-s − 0.120·69-s − 0.593·71-s + 1.75·73-s + 0.577·75-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
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 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−16.80835305113360, −16.66587056246160, −15.66983247643006, −15.25931902940550, −14.45720536121246, −14.05659894524625, −13.63125940506927, −12.72707853213356, −11.99807067321074, −11.71632357317441, −11.19155123981359, −10.60922132016348, −9.875421486732918, −9.019228761680257, −8.744261333362011, −8.042019150964037, −7.096270072234887, −6.533002590536844, −5.997000916113788, −5.331277079550149, −4.580131821802789, −3.725409319125754, −3.245729740580509, −1.869001162576256, −1.276997807937623, 0, 1.276997807937623, 1.869001162576256, 3.245729740580509, 3.725409319125754, 4.580131821802789, 5.331277079550149, 5.997000916113788, 6.533002590536844, 7.096270072234887, 8.042019150964037, 8.744261333362011, 9.019228761680257, 9.875421486732918, 10.60922132016348, 11.19155123981359, 11.71632357317441, 11.99807067321074, 12.72707853213356, 13.63125940506927, 14.05659894524625, 14.45720536121246, 15.25931902940550, 15.66983247643006, 16.66587056246160, 16.80835305113360

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