Properties

Label 2-1805-1.1-c1-0-37
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  − 1.68·2-s + 3.25·3-s + 0.834·4-s − 5-s − 5.47·6-s − 0.548·7-s + 1.96·8-s + 7.57·9-s + 1.68·10-s + 0.331·11-s + 2.71·12-s − 4.82·13-s + 0.923·14-s − 3.25·15-s − 4.97·16-s + 5.28·17-s − 12.7·18-s − 0.834·20-s − 1.78·21-s − 0.557·22-s − 1.11·23-s + 6.38·24-s + 25-s + 8.11·26-s + 14.8·27-s − 0.457·28-s + 3.20·29-s + ⋯
L(s)  = 1  − 1.19·2-s + 1.87·3-s + 0.417·4-s − 0.447·5-s − 2.23·6-s − 0.207·7-s + 0.693·8-s + 2.52·9-s + 0.532·10-s + 0.0998·11-s + 0.783·12-s − 1.33·13-s + 0.246·14-s − 0.839·15-s − 1.24·16-s + 1.28·17-s − 3.00·18-s − 0.186·20-s − 0.389·21-s − 0.118·22-s − 0.231·23-s + 1.30·24-s + 0.200·25-s + 1.59·26-s + 2.86·27-s − 0.0864·28-s + 0.595·29-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

\(L(1)\) \(\approx\) \(1.655553084\)
\(L(\frac12)\) \(\approx\) \(1.655553084\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.68T + 2T^{2} \)
3 \( 1 - 3.25T + 3T^{2} \)
7 \( 1 + 0.548T + 7T^{2} \)
11 \( 1 - 0.331T + 11T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 - 5.28T + 17T^{2} \)
23 \( 1 + 1.11T + 23T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 - 6.02T + 31T^{2} \)
37 \( 1 - 6.67T + 37T^{2} \)
41 \( 1 - 7.91T + 41T^{2} \)
43 \( 1 - 1.63T + 43T^{2} \)
47 \( 1 + 4.60T + 47T^{2} \)
53 \( 1 + 6.72T + 53T^{2} \)
59 \( 1 - 6.80T + 59T^{2} \)
61 \( 1 - 0.679T + 61T^{2} \)
67 \( 1 - 7.77T + 67T^{2} \)
71 \( 1 - 1.08T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 - 1.40T + 79T^{2} \)
83 \( 1 + 0.854T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 - 7.02T + 97T^{2} \)
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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

−9.317102391230380052103692129933, −8.381519616324275567054959121244, −7.921828995348341219218046755991, −7.49734905915166939247796489327, −6.63492174840907029993106649661, −4.88034335292642920764473774470, −4.08862722932527688584314392463, −3.05215522703167314616269811377, −2.25593230279398055999845908699, −1.00643514399306799469999581845, 1.00643514399306799469999581845, 2.25593230279398055999845908699, 3.05215522703167314616269811377, 4.08862722932527688584314392463, 4.88034335292642920764473774470, 6.63492174840907029993106649661, 7.49734905915166939247796489327, 7.921828995348341219218046755991, 8.381519616324275567054959121244, 9.317102391230380052103692129933

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