Properties

Label 2-1805-1.1-c1-0-65
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.820·2-s − 2.32·3-s − 1.32·4-s + 5-s − 1.90·6-s − 0.561·7-s − 2.72·8-s + 2.41·9-s + 0.820·10-s − 0.111·11-s + 3.08·12-s + 6.89·13-s − 0.460·14-s − 2.32·15-s + 0.415·16-s − 3.81·17-s + 1.98·18-s − 1.32·20-s + 1.30·21-s − 0.0911·22-s − 4.04·23-s + 6.35·24-s + 25-s + 5.65·26-s + 1.35·27-s + 0.745·28-s + 9.29·29-s + ⋯
L(s)  = 1  + 0.580·2-s − 1.34·3-s − 0.663·4-s + 0.447·5-s − 0.779·6-s − 0.212·7-s − 0.964·8-s + 0.805·9-s + 0.259·10-s − 0.0334·11-s + 0.891·12-s + 1.91·13-s − 0.123·14-s − 0.600·15-s + 0.103·16-s − 0.924·17-s + 0.467·18-s − 0.296·20-s + 0.285·21-s − 0.0194·22-s − 0.842·23-s + 1.29·24-s + 0.200·25-s + 1.10·26-s + 0.261·27-s + 0.140·28-s + 1.72·29-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: $\chi_{1805} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
19 \( 1 \)
good2 \( 1 - 0.820T + 2T^{2} \)
3 \( 1 + 2.32T + 3T^{2} \)
7 \( 1 + 0.561T + 7T^{2} \)
11 \( 1 + 0.111T + 11T^{2} \)
13 \( 1 - 6.89T + 13T^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
23 \( 1 + 4.04T + 23T^{2} \)
29 \( 1 - 9.29T + 29T^{2} \)
31 \( 1 + 4.18T + 31T^{2} \)
37 \( 1 - 2.68T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 + 9.63T + 43T^{2} \)
47 \( 1 - 2.12T + 47T^{2} \)
53 \( 1 + 5.74T + 53T^{2} \)
59 \( 1 - 7.89T + 59T^{2} \)
61 \( 1 - 5.56T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 6.64T + 71T^{2} \)
73 \( 1 + 8.45T + 73T^{2} \)
79 \( 1 + 6.27T + 79T^{2} \)
83 \( 1 + 8.16T + 83T^{2} \)
89 \( 1 - 1.16T + 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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

−8.727390608412425507723086983089, −8.394665759903286237497965982255, −6.73437144441975477482652207910, −6.26082470991179082995167469003, −5.67789132021935583309302244838, −4.86966431752158648384806420625, −4.10708829601966542975994261094, −3.10581063877843075820460278495, −1.38814385731444096100905722486, 0, 1.38814385731444096100905722486, 3.10581063877843075820460278495, 4.10708829601966542975994261094, 4.86966431752158648384806420625, 5.67789132021935583309302244838, 6.26082470991179082995167469003, 6.73437144441975477482652207910, 8.394665759903286237497965982255, 8.727390608412425507723086983089

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