Properties

Label 2-1805-1.1-c1-0-109
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  + 2.62·2-s − 0.928·3-s + 4.87·4-s − 5-s − 2.43·6-s − 3.83·7-s + 7.54·8-s − 2.13·9-s − 2.62·10-s − 3.26·11-s − 4.52·12-s − 0.364·13-s − 10.0·14-s + 0.928·15-s + 10.0·16-s − 0.684·17-s − 5.60·18-s − 4.87·20-s + 3.55·21-s − 8.56·22-s − 9.36·23-s − 7.00·24-s + 25-s − 0.954·26-s + 4.77·27-s − 18.6·28-s + 1.53·29-s + ⋯
L(s)  = 1  + 1.85·2-s − 0.536·3-s + 2.43·4-s − 0.447·5-s − 0.994·6-s − 1.44·7-s + 2.66·8-s − 0.712·9-s − 0.829·10-s − 0.985·11-s − 1.30·12-s − 0.100·13-s − 2.68·14-s + 0.239·15-s + 2.50·16-s − 0.165·17-s − 1.32·18-s − 1.09·20-s + 0.776·21-s − 1.82·22-s − 1.95·23-s − 1.42·24-s + 0.200·25-s − 0.187·26-s + 0.918·27-s − 3.53·28-s + 0.285·29-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: \(1\)
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 \( 1 - 2.62T + 2T^{2} \)
3 \( 1 + 0.928T + 3T^{2} \)
7 \( 1 + 3.83T + 7T^{2} \)
11 \( 1 + 3.26T + 11T^{2} \)
13 \( 1 + 0.364T + 13T^{2} \)
17 \( 1 + 0.684T + 17T^{2} \)
23 \( 1 + 9.36T + 23T^{2} \)
29 \( 1 - 1.53T + 29T^{2} \)
31 \( 1 + 2.44T + 31T^{2} \)
37 \( 1 - 0.163T + 37T^{2} \)
41 \( 1 + 7.37T + 41T^{2} \)
43 \( 1 - 2.58T + 43T^{2} \)
47 \( 1 - 8.79T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + 7.78T + 59T^{2} \)
61 \( 1 - 2.41T + 61T^{2} \)
67 \( 1 + 9.55T + 67T^{2} \)
71 \( 1 + 5.12T + 71T^{2} \)
73 \( 1 + 7.88T + 73T^{2} \)
79 \( 1 + 9.32T + 79T^{2} \)
83 \( 1 - 0.729T + 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + 7.54T + 97T^{2} \)
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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

−8.783499835223777633426298730566, −7.67295539213433502940224259342, −6.93960702924550343661921804465, −5.99711294919372441544580679373, −5.79758209265530482286314978517, −4.77037317854026560524039496588, −3.87286781261623783483957892016, −3.10319389933574561654846941743, −2.34078253538495758748695285855, 0, 2.34078253538495758748695285855, 3.10319389933574561654846941743, 3.87286781261623783483957892016, 4.77037317854026560524039496588, 5.79758209265530482286314978517, 5.99711294919372441544580679373, 6.93960702924550343661921804465, 7.67295539213433502940224259342, 8.783499835223777633426298730566

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