Properties

Label 2-1805-1.1-c1-0-105
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  + 1.75·2-s + 0.0856·3-s + 1.08·4-s + 5-s + 0.150·6-s − 1.86·7-s − 1.60·8-s − 2.99·9-s + 1.75·10-s − 1.22·11-s + 0.0929·12-s − 2.40·13-s − 3.26·14-s + 0.0856·15-s − 4.99·16-s + 6.05·17-s − 5.25·18-s + 1.08·20-s − 0.159·21-s − 2.15·22-s − 7.20·23-s − 0.137·24-s + 25-s − 4.22·26-s − 0.513·27-s − 2.02·28-s + 1.28·29-s + ⋯
L(s)  = 1  + 1.24·2-s + 0.0494·3-s + 0.542·4-s + 0.447·5-s + 0.0614·6-s − 0.703·7-s − 0.567·8-s − 0.997·9-s + 0.555·10-s − 0.369·11-s + 0.0268·12-s − 0.667·13-s − 0.873·14-s + 0.0221·15-s − 1.24·16-s + 1.46·17-s − 1.23·18-s + 0.242·20-s − 0.0347·21-s − 0.458·22-s − 1.50·23-s − 0.0280·24-s + 0.200·25-s − 0.829·26-s − 0.0987·27-s − 0.381·28-s + 0.238·29-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 \( 1 - 1.75T + 2T^{2} \)
3 \( 1 - 0.0856T + 3T^{2} \)
7 \( 1 + 1.86T + 7T^{2} \)
11 \( 1 + 1.22T + 11T^{2} \)
13 \( 1 + 2.40T + 13T^{2} \)
17 \( 1 - 6.05T + 17T^{2} \)
23 \( 1 + 7.20T + 23T^{2} \)
29 \( 1 - 1.28T + 29T^{2} \)
31 \( 1 + 5.87T + 31T^{2} \)
37 \( 1 + 1.59T + 37T^{2} \)
41 \( 1 + 9.72T + 41T^{2} \)
43 \( 1 + 0.697T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 - 9.40T + 59T^{2} \)
61 \( 1 - 3.73T + 61T^{2} \)
67 \( 1 - 7.41T + 67T^{2} \)
71 \( 1 + 8.51T + 71T^{2} \)
73 \( 1 + 6.21T + 73T^{2} \)
79 \( 1 + 7.39T + 79T^{2} \)
83 \( 1 - 0.0135T + 83T^{2} \)
89 \( 1 + 1.25T + 89T^{2} \)
97 \( 1 - 7.97T + 97T^{2} \)
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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

−8.871831568507156595954593274596, −8.087420512136169007297023690388, −7.02240182901016060334251025893, −6.12031500498692171299919701468, −5.55493649976042875383061307124, −4.96298457855662896642007588427, −3.67953200170138527747880619613, −3.12967804780164720681069809144, −2.14910618556348512450075785163, 0, 2.14910618556348512450075785163, 3.12967804780164720681069809144, 3.67953200170138527747880619613, 4.96298457855662896642007588427, 5.55493649976042875383061307124, 6.12031500498692171299919701468, 7.02240182901016060334251025893, 8.087420512136169007297023690388, 8.871831568507156595954593274596

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