Properties

Label 2-1805-1.1-c1-0-32
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  − 0.503·2-s − 3.01·3-s − 1.74·4-s + 5-s + 1.51·6-s + 2.48·7-s + 1.88·8-s + 6.06·9-s − 0.503·10-s + 4.58·11-s + 5.25·12-s + 6.06·13-s − 1.25·14-s − 3.01·15-s + 2.54·16-s + 1.83·17-s − 3.05·18-s − 1.74·20-s − 7.49·21-s − 2.30·22-s + 0.125·23-s − 5.67·24-s + 25-s − 3.05·26-s − 9.21·27-s − 4.34·28-s − 3.79·29-s + ⋯
L(s)  = 1  − 0.356·2-s − 1.73·3-s − 0.873·4-s + 0.447·5-s + 0.618·6-s + 0.941·7-s + 0.666·8-s + 2.02·9-s − 0.159·10-s + 1.38·11-s + 1.51·12-s + 1.68·13-s − 0.335·14-s − 0.777·15-s + 0.635·16-s + 0.444·17-s − 0.719·18-s − 0.390·20-s − 1.63·21-s − 0.491·22-s + 0.0262·23-s − 1.15·24-s + 0.200·25-s − 0.598·26-s − 1.77·27-s − 0.821·28-s − 0.703·29-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

\(L(1)\) \(\approx\) \(1.020390655\)
\(L(\frac12)\) \(\approx\) \(1.020390655\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.503T + 2T^{2} \)
3 \( 1 + 3.01T + 3T^{2} \)
7 \( 1 - 2.48T + 7T^{2} \)
11 \( 1 - 4.58T + 11T^{2} \)
13 \( 1 - 6.06T + 13T^{2} \)
17 \( 1 - 1.83T + 17T^{2} \)
23 \( 1 - 0.125T + 23T^{2} \)
29 \( 1 + 3.79T + 29T^{2} \)
31 \( 1 - 8.09T + 31T^{2} \)
37 \( 1 + 1.44T + 37T^{2} \)
41 \( 1 + 3.02T + 41T^{2} \)
43 \( 1 - 8.78T + 43T^{2} \)
47 \( 1 - 5.70T + 47T^{2} \)
53 \( 1 - 9.24T + 53T^{2} \)
59 \( 1 + 7.06T + 59T^{2} \)
61 \( 1 + 8.41T + 61T^{2} \)
67 \( 1 - 6.08T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 - 5.23T + 79T^{2} \)
83 \( 1 - 5.03T + 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + 2.49T + 97T^{2} \)
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

−9.289772796560999034230062174094, −8.636505754424566468860330425190, −7.71019839873704282763957078713, −6.63958328658564291861369565005, −5.97213569413620777241214963554, −5.33657565508549433675573515076, −4.44413807339140922425978076742, −3.83171457425077008087788177568, −1.45777694367893897722316644020, −0.964360186067930435772160390631, 0.964360186067930435772160390631, 1.45777694367893897722316644020, 3.83171457425077008087788177568, 4.44413807339140922425978076742, 5.33657565508549433675573515076, 5.97213569413620777241214963554, 6.63958328658564291861369565005, 7.71019839873704282763957078713, 8.636505754424566468860330425190, 9.289772796560999034230062174094

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