Properties

Label 2-304-1.1-c3-0-14
Degree $2$
Conductor $304$
Sign $-1$
Analytic cond. $17.9365$
Root an. cond. $4.23516$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8.28·3-s + 2.38·5-s − 5.83·7-s + 41.6·9-s + 7.33·11-s + 55.6·13-s − 19.7·15-s − 10.0·17-s + 19·19-s + 48.3·21-s + 9.26·23-s − 119.·25-s − 121.·27-s − 83.9·29-s − 202.·31-s − 60.7·33-s − 13.9·35-s + 95.2·37-s − 460.·39-s − 25.9·41-s + 119.·43-s + 99.3·45-s − 467.·47-s − 308.·49-s + 83.1·51-s − 764.·53-s + 17.4·55-s + ⋯
L(s)  = 1  − 1.59·3-s + 0.213·5-s − 0.315·7-s + 1.54·9-s + 0.201·11-s + 1.18·13-s − 0.340·15-s − 0.143·17-s + 0.229·19-s + 0.502·21-s + 0.0839·23-s − 0.954·25-s − 0.866·27-s − 0.537·29-s − 1.17·31-s − 0.320·33-s − 0.0671·35-s + 0.423·37-s − 1.89·39-s − 0.0990·41-s + 0.424·43-s + 0.329·45-s − 1.45·47-s − 0.900·49-s + 0.228·51-s − 1.98·53-s + 0.0428·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-1$
Analytic conductor: \(17.9365\)
Root analytic conductor: \(4.23516\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 304,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - 19T \)
good3 \( 1 + 8.28T + 27T^{2} \)
5 \( 1 - 2.38T + 125T^{2} \)
7 \( 1 + 5.83T + 343T^{2} \)
11 \( 1 - 7.33T + 1.33e3T^{2} \)
13 \( 1 - 55.6T + 2.19e3T^{2} \)
17 \( 1 + 10.0T + 4.91e3T^{2} \)
23 \( 1 - 9.26T + 1.21e4T^{2} \)
29 \( 1 + 83.9T + 2.43e4T^{2} \)
31 \( 1 + 202.T + 2.97e4T^{2} \)
37 \( 1 - 95.2T + 5.06e4T^{2} \)
41 \( 1 + 25.9T + 6.89e4T^{2} \)
43 \( 1 - 119.T + 7.95e4T^{2} \)
47 \( 1 + 467.T + 1.03e5T^{2} \)
53 \( 1 + 764.T + 1.48e5T^{2} \)
59 \( 1 + 69.1T + 2.05e5T^{2} \)
61 \( 1 + 398.T + 2.26e5T^{2} \)
67 \( 1 - 243.T + 3.00e5T^{2} \)
71 \( 1 - 781.T + 3.57e5T^{2} \)
73 \( 1 + 711.T + 3.89e5T^{2} \)
79 \( 1 + 723.T + 4.93e5T^{2} \)
83 \( 1 - 1.22e3T + 5.71e5T^{2} \)
89 \( 1 + 653.T + 7.04e5T^{2} \)
97 \( 1 - 1.69e3T + 9.12e5T^{2} \)
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

−11.11478154261477393370091807576, −10.07631081111263262342216007006, −9.145943323624515733377138700284, −7.72927760168752107329528784030, −6.48096883962091282516400275652, −5.96353221531555567172647901773, −4.92511699806790761660551946803, −3.63276152953702285859411546318, −1.49090226129081519846287847615, 0, 1.49090226129081519846287847615, 3.63276152953702285859411546318, 4.92511699806790761660551946803, 5.96353221531555567172647901773, 6.48096883962091282516400275652, 7.72927760168752107329528784030, 9.145943323624515733377138700284, 10.07631081111263262342216007006, 11.11478154261477393370091807576

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