Properties

Label 2-1805-1.1-c1-0-27
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.46·2-s − 1.51·3-s + 0.138·4-s − 5-s − 2.20·6-s + 4.07·7-s − 2.72·8-s − 0.718·9-s − 1.46·10-s − 0.621·11-s − 0.208·12-s + 5.09·13-s + 5.95·14-s + 1.51·15-s − 4.25·16-s − 0.266·17-s − 1.05·18-s − 0.138·20-s − 6.14·21-s − 0.908·22-s − 5.84·23-s + 4.11·24-s + 25-s + 7.44·26-s + 5.61·27-s + 0.563·28-s − 4.07·29-s + ⋯
L(s)  = 1  + 1.03·2-s − 0.872·3-s + 0.0691·4-s − 0.447·5-s − 0.901·6-s + 1.53·7-s − 0.962·8-s − 0.239·9-s − 0.462·10-s − 0.187·11-s − 0.0603·12-s + 1.41·13-s + 1.59·14-s + 0.389·15-s − 1.06·16-s − 0.0646·17-s − 0.247·18-s − 0.0309·20-s − 1.34·21-s − 0.193·22-s − 1.21·23-s + 0.839·24-s + 0.200·25-s + 1.46·26-s + 1.08·27-s + 0.106·28-s − 0.757·29-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

\(L(1)\) \(\approx\) \(1.949279869\)
\(L(\frac12)\) \(\approx\) \(1.949279869\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.46T + 2T^{2} \)
3 \( 1 + 1.51T + 3T^{2} \)
7 \( 1 - 4.07T + 7T^{2} \)
11 \( 1 + 0.621T + 11T^{2} \)
13 \( 1 - 5.09T + 13T^{2} \)
17 \( 1 + 0.266T + 17T^{2} \)
23 \( 1 + 5.84T + 23T^{2} \)
29 \( 1 + 4.07T + 29T^{2} \)
31 \( 1 - 6.48T + 31T^{2} \)
37 \( 1 - 8.83T + 37T^{2} \)
41 \( 1 - 4.48T + 41T^{2} \)
43 \( 1 + 1.80T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 2.52T + 53T^{2} \)
59 \( 1 + 0.890T + 59T^{2} \)
61 \( 1 - 2.16T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 - 1.64T + 71T^{2} \)
73 \( 1 - 1.79T + 73T^{2} \)
79 \( 1 + 5.14T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 0.000747T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.103133210556520497636768980348, −8.335647491255021348953351132534, −7.80021279753426929653795285228, −6.44924788672026532460191486931, −5.83381837801622388050469992896, −5.19726550866396637525091467196, −4.38062898876103329621573459665, −3.79513057237344794271858835850, −2.43384355747603962787730236606, −0.875883791869178934438611308451, 0.875883791869178934438611308451, 2.43384355747603962787730236606, 3.79513057237344794271858835850, 4.38062898876103329621573459665, 5.19726550866396637525091467196, 5.83381837801622388050469992896, 6.44924788672026532460191486931, 7.80021279753426929653795285228, 8.335647491255021348953351132534, 9.103133210556520497636768980348

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