Properties

Label 2-1805-1.1-c1-0-99
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  + 3.07·3-s − 2·4-s − 5-s − 0.618·7-s + 6.47·9-s − 5.85·11-s − 6.15·12-s − 3.07·13-s − 3.07·15-s + 4·16-s − 2.85·17-s + 2·20-s − 1.90·21-s + 5.47·23-s + 25-s + 10.6·27-s + 1.23·28-s − 3.80·29-s − 1.62·31-s − 18.0·33-s + 0.618·35-s − 12.9·36-s − 8.33·37-s − 9.47·39-s − 11.5·41-s − 7.38·43-s + 11.7·44-s + ⋯
L(s)  = 1  + 1.77·3-s − 4-s − 0.447·5-s − 0.233·7-s + 2.15·9-s − 1.76·11-s − 1.77·12-s − 0.853·13-s − 0.794·15-s + 16-s − 0.692·17-s + 0.447·20-s − 0.415·21-s + 1.14·23-s + 0.200·25-s + 2.05·27-s + 0.233·28-s − 0.706·29-s − 0.291·31-s − 3.13·33-s + 0.104·35-s − 2.15·36-s − 1.37·37-s − 1.51·39-s − 1.80·41-s − 1.12·43-s + 1.76·44-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 \( 1 + 2T^{2} \)
3 \( 1 - 3.07T + 3T^{2} \)
7 \( 1 + 0.618T + 7T^{2} \)
11 \( 1 + 5.85T + 11T^{2} \)
13 \( 1 + 3.07T + 13T^{2} \)
17 \( 1 + 2.85T + 17T^{2} \)
23 \( 1 - 5.47T + 23T^{2} \)
29 \( 1 + 3.80T + 29T^{2} \)
31 \( 1 + 1.62T + 31T^{2} \)
37 \( 1 + 8.33T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 + 7.38T + 43T^{2} \)
47 \( 1 - 4.70T + 47T^{2} \)
53 \( 1 + 5.15T + 53T^{2} \)
59 \( 1 - 4.25T + 59T^{2} \)
61 \( 1 + 7.23T + 61T^{2} \)
67 \( 1 - 7.33T + 67T^{2} \)
71 \( 1 - 0.171T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 8.22T + 89T^{2} \)
97 \( 1 + 8.33T + 97T^{2} \)
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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

−8.766507504285431811317235901074, −8.233216170048475047319018157235, −7.59801988034984075500451516351, −6.91591456807867376754611517098, −5.17802306067682585821016884206, −4.72897109025186701158251897407, −3.55586122555851354197136028429, −3.03075989187154571245610161709, −1.97247302592756378320630139447, 0, 1.97247302592756378320630139447, 3.03075989187154571245610161709, 3.55586122555851354197136028429, 4.72897109025186701158251897407, 5.17802306067682585821016884206, 6.91591456807867376754611517098, 7.59801988034984075500451516351, 8.233216170048475047319018157235, 8.766507504285431811317235901074

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