Properties

Label 2-304-19.18-c4-0-7
Degree $2$
Conductor $304$
Sign $1$
Analytic cond. $31.4244$
Root an. cond. $5.60575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 49.4·5-s − 93.1·7-s + 81·9-s − 173.·11-s − 219.·17-s − 361·19-s + 158·23-s + 1.82e3·25-s + 4.60e3·35-s + 800.·43-s − 4.00e3·45-s − 3.07e3·47-s + 6.27e3·49-s + 8.56e3·55-s − 7.41e3·61-s − 7.54e3·63-s + 1.90e3·73-s + 1.61e4·77-s + 6.56e3·81-s + 5.67e3·83-s + 1.08e4·85-s + 1.78e4·95-s − 1.40e4·99-s − 9.99e3·101-s − 7.81e3·115-s + 2.04e4·119-s + ⋯
L(s)  = 1  − 1.97·5-s − 1.90·7-s + 9-s − 1.43·11-s − 0.760·17-s − 19-s + 0.298·23-s + 2.91·25-s + 3.76·35-s + 0.433·43-s − 1.97·45-s − 1.39·47-s + 2.61·49-s + 2.83·55-s − 1.99·61-s − 1.90·63-s + 0.356·73-s + 2.71·77-s + 81-s + 0.824·83-s + 1.50·85-s + 1.97·95-s − 1.43·99-s − 0.980·101-s − 0.591·115-s + 1.44·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $1$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (113, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.3189813896\)
\(L(\frac12)\) \(\approx\) \(0.3189813896\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + 361T \)
good3 \( 1 - 81T^{2} \)
5 \( 1 + 49.4T + 625T^{2} \)
7 \( 1 + 93.1T + 2.40e3T^{2} \)
11 \( 1 + 173.T + 1.46e4T^{2} \)
13 \( 1 - 2.85e4T^{2} \)
17 \( 1 + 219.T + 8.35e4T^{2} \)
23 \( 1 - 158T + 2.79e5T^{2} \)
29 \( 1 - 7.07e5T^{2} \)
31 \( 1 - 9.23e5T^{2} \)
37 \( 1 - 1.87e6T^{2} \)
41 \( 1 - 2.82e6T^{2} \)
43 \( 1 - 800.T + 3.41e6T^{2} \)
47 \( 1 + 3.07e3T + 4.87e6T^{2} \)
53 \( 1 - 7.89e6T^{2} \)
59 \( 1 - 1.21e7T^{2} \)
61 \( 1 + 7.41e3T + 1.38e7T^{2} \)
67 \( 1 - 2.01e7T^{2} \)
71 \( 1 - 2.54e7T^{2} \)
73 \( 1 - 1.90e3T + 2.83e7T^{2} \)
79 \( 1 - 3.89e7T^{2} \)
83 \( 1 - 5.67e3T + 4.74e7T^{2} \)
89 \( 1 - 6.27e7T^{2} \)
97 \( 1 - 8.85e7T^{2} \)
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

−10.95240219815696386526109384201, −10.32115537757384892687341971643, −9.158601694087756824725542624961, −8.065892195228873796997326352875, −7.22064065071941939568119767965, −6.46842371030224318748053220339, −4.70829232359955676675288697186, −3.78434703301554825974946216427, −2.83370906936667219368000080827, −0.32603285767085400083839343168, 0.32603285767085400083839343168, 2.83370906936667219368000080827, 3.78434703301554825974946216427, 4.70829232359955676675288697186, 6.46842371030224318748053220339, 7.22064065071941939568119767965, 8.065892195228873796997326352875, 9.158601694087756824725542624961, 10.32115537757384892687341971643, 10.95240219815696386526109384201

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