Properties

Label 2-1805-1.1-c1-0-87
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.43·2-s + 2.94·3-s + 3.94·4-s + 5-s − 7.18·6-s − 3.82·7-s − 4.74·8-s + 5.67·9-s − 2.43·10-s − 2.12·11-s + 11.6·12-s − 3.65·13-s + 9.31·14-s + 2.94·15-s + 3.67·16-s − 3.04·17-s − 13.8·18-s + 3.94·20-s − 11.2·21-s + 5.18·22-s − 4.81·23-s − 13.9·24-s + 25-s + 8.91·26-s + 7.87·27-s − 15.0·28-s + 6.03·29-s + ⋯
L(s)  = 1  − 1.72·2-s + 1.70·3-s + 1.97·4-s + 0.447·5-s − 2.93·6-s − 1.44·7-s − 1.67·8-s + 1.89·9-s − 0.771·10-s − 0.640·11-s + 3.35·12-s − 1.01·13-s + 2.48·14-s + 0.760·15-s + 0.918·16-s − 0.737·17-s − 3.26·18-s + 0.882·20-s − 2.45·21-s + 1.10·22-s − 1.00·23-s − 2.85·24-s + 0.200·25-s + 1.74·26-s + 1.51·27-s − 2.84·28-s + 1.12·29-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 \( 1 + 2.43T + 2T^{2} \)
3 \( 1 - 2.94T + 3T^{2} \)
7 \( 1 + 3.82T + 7T^{2} \)
11 \( 1 + 2.12T + 11T^{2} \)
13 \( 1 + 3.65T + 13T^{2} \)
17 \( 1 + 3.04T + 17T^{2} \)
23 \( 1 + 4.81T + 23T^{2} \)
29 \( 1 - 6.03T + 29T^{2} \)
31 \( 1 - 3.57T + 31T^{2} \)
37 \( 1 - 3.93T + 37T^{2} \)
41 \( 1 + 7.60T + 41T^{2} \)
43 \( 1 + 5.60T + 43T^{2} \)
47 \( 1 + 8.41T + 47T^{2} \)
53 \( 1 - 4.80T + 53T^{2} \)
59 \( 1 + 5.13T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 + 5.38T + 67T^{2} \)
71 \( 1 + 0.123T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 + 4.25T + 79T^{2} \)
83 \( 1 - 4.39T + 83T^{2} \)
89 \( 1 + 0.0772T + 89T^{2} \)
97 \( 1 + 18.0T + 97T^{2} \)
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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

−8.939294838340416080732404248124, −8.300958671607449761102399638476, −7.64049767910731381906522320492, −6.87949053178501435282410532768, −6.24256115155707740896494620499, −4.55188130547868824581503151623, −3.11010337654967640435408234786, −2.64652863544610167503576718447, −1.76492921063241349373842219227, 0, 1.76492921063241349373842219227, 2.64652863544610167503576718447, 3.11010337654967640435408234786, 4.55188130547868824581503151623, 6.24256115155707740896494620499, 6.87949053178501435282410532768, 7.64049767910731381906522320492, 8.300958671607449761102399638476, 8.939294838340416080732404248124

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