Properties

Label 2-1805-1.1-c1-0-56
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  + 1.25·2-s + 1.78·3-s − 0.414·4-s + 5-s + 2.24·6-s − 0.828·7-s − 3.04·8-s + 0.171·9-s + 1.25·10-s + 2·11-s − 0.737·12-s + 4.29·13-s − 1.04·14-s + 1.78·15-s − 3·16-s + 7.65·17-s + 0.216·18-s − 0.414·20-s − 1.47·21-s + 2.51·22-s − 0.828·23-s − 5.41·24-s + 25-s + 5.41·26-s − 5.03·27-s + 0.343·28-s + 8.59·29-s + ⋯
L(s)  = 1  + 0.890·2-s + 1.02·3-s − 0.207·4-s + 0.447·5-s + 0.915·6-s − 0.313·7-s − 1.07·8-s + 0.0571·9-s + 0.398·10-s + 0.603·11-s − 0.212·12-s + 1.19·13-s − 0.278·14-s + 0.459·15-s − 0.750·16-s + 1.85·17-s + 0.0509·18-s − 0.0926·20-s − 0.321·21-s + 0.536·22-s − 0.172·23-s − 1.10·24-s + 0.200·25-s + 1.06·26-s − 0.969·27-s + 0.0648·28-s + 1.59·29-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: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.760837977\)
\(L(\frac12)\) \(\approx\) \(3.760837977\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.25T + 2T^{2} \)
3 \( 1 - 1.78T + 3T^{2} \)
7 \( 1 + 0.828T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 4.29T + 13T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
23 \( 1 + 0.828T + 23T^{2} \)
29 \( 1 - 8.59T + 29T^{2} \)
31 \( 1 - 3.56T + 31T^{2} \)
37 \( 1 - 0.737T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4.82T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 7.86T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 - 6.81T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 7.12T + 79T^{2} \)
83 \( 1 - 8.82T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 - 4.29T + 97T^{2} \)
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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

−9.311261933989550705112117303709, −8.436520714630898826840724303035, −7.986356977192196497805234799319, −6.58899569083391258859789035322, −6.00256137811455663746926578109, −5.18529548245026991869809452313, −4.06693877657075806410018931849, −3.36398341508515669174902282783, −2.76780591179750605253961810722, −1.23339406749050078765882619428, 1.23339406749050078765882619428, 2.76780591179750605253961810722, 3.36398341508515669174902282783, 4.06693877657075806410018931849, 5.18529548245026991869809452313, 6.00256137811455663746926578109, 6.58899569083391258859789035322, 7.986356977192196497805234799319, 8.436520714630898826840724303035, 9.311261933989550705112117303709

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