Properties

Label 2-1805-1.1-c1-0-31
Degree $2$
Conductor $1805$
Sign $1$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.09·2-s − 0.379·3-s − 0.795·4-s + 5-s − 0.416·6-s + 1.89·7-s − 3.06·8-s − 2.85·9-s + 1.09·10-s + 0.134·11-s + 0.301·12-s + 3.51·13-s + 2.07·14-s − 0.379·15-s − 1.77·16-s − 1.66·17-s − 3.13·18-s − 0.795·20-s − 0.718·21-s + 0.147·22-s + 5.36·23-s + 1.16·24-s + 25-s + 3.85·26-s + 2.22·27-s − 1.50·28-s + 4.97·29-s + ⋯
L(s)  = 1  + 0.776·2-s − 0.219·3-s − 0.397·4-s + 0.447·5-s − 0.169·6-s + 0.715·7-s − 1.08·8-s − 0.952·9-s + 0.347·10-s + 0.0405·11-s + 0.0871·12-s + 0.974·13-s + 0.555·14-s − 0.0979·15-s − 0.443·16-s − 0.402·17-s − 0.738·18-s − 0.177·20-s − 0.156·21-s + 0.0314·22-s + 1.11·23-s + 0.237·24-s + 0.200·25-s + 0.756·26-s + 0.427·27-s − 0.284·28-s + 0.923·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.228926084\)
\(L(\frac12)\) \(\approx\) \(2.228926084\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
19 \( 1 \)
good2 \( 1 - 1.09T + 2T^{2} \)
3 \( 1 + 0.379T + 3T^{2} \)
7 \( 1 - 1.89T + 7T^{2} \)
11 \( 1 - 0.134T + 11T^{2} \)
13 \( 1 - 3.51T + 13T^{2} \)
17 \( 1 + 1.66T + 17T^{2} \)
23 \( 1 - 5.36T + 23T^{2} \)
29 \( 1 - 4.97T + 29T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 + 1.69T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 8.50T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + 0.264T + 53T^{2} \)
59 \( 1 + 6.89T + 59T^{2} \)
61 \( 1 - 9.17T + 61T^{2} \)
67 \( 1 + 2.95T + 67T^{2} \)
71 \( 1 - 1.32T + 71T^{2} \)
73 \( 1 + 6.34T + 73T^{2} \)
79 \( 1 + 1.46T + 79T^{2} \)
83 \( 1 - 7.44T + 83T^{2} \)
89 \( 1 - 9.73T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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

−9.002945506606027377132293083017, −8.688452569833422037957568778835, −7.80217009315196699771784659324, −6.41652300670887493039467820260, −6.03087779086326366838430068012, −5.05780752010583385045590106771, −4.56684970062580303927838719419, −3.41092448632297249398630390656, −2.53645489756928128075821024329, −0.953279651893298734050298226696, 0.953279651893298734050298226696, 2.53645489756928128075821024329, 3.41092448632297249398630390656, 4.56684970062580303927838719419, 5.05780752010583385045590106771, 6.03087779086326366838430068012, 6.41652300670887493039467820260, 7.80217009315196699771784659324, 8.688452569833422037957568778835, 9.002945506606027377132293083017

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