Properties

Label 2-1805-1.1-c1-0-54
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  − 1.25·2-s − 3.01·3-s − 0.420·4-s + 5-s + 3.78·6-s + 3.72·7-s + 3.04·8-s + 6.07·9-s − 1.25·10-s − 3.35·11-s + 1.26·12-s − 4.84·13-s − 4.68·14-s − 3.01·15-s − 2.98·16-s − 2.67·17-s − 7.63·18-s − 0.420·20-s − 11.2·21-s + 4.21·22-s + 1.87·23-s − 9.16·24-s + 25-s + 6.08·26-s − 9.25·27-s − 1.56·28-s + 5.25·29-s + ⋯
L(s)  = 1  − 0.888·2-s − 1.73·3-s − 0.210·4-s + 0.447·5-s + 1.54·6-s + 1.40·7-s + 1.07·8-s + 2.02·9-s − 0.397·10-s − 1.01·11-s + 0.365·12-s − 1.34·13-s − 1.25·14-s − 0.777·15-s − 0.745·16-s − 0.648·17-s − 1.79·18-s − 0.0939·20-s − 2.44·21-s + 0.899·22-s + 0.391·23-s − 1.87·24-s + 0.200·25-s + 1.19·26-s − 1.78·27-s − 0.295·28-s + 0.975·29-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 \( 1 + 1.25T + 2T^{2} \)
3 \( 1 + 3.01T + 3T^{2} \)
7 \( 1 - 3.72T + 7T^{2} \)
11 \( 1 + 3.35T + 11T^{2} \)
13 \( 1 + 4.84T + 13T^{2} \)
17 \( 1 + 2.67T + 17T^{2} \)
23 \( 1 - 1.87T + 23T^{2} \)
29 \( 1 - 5.25T + 29T^{2} \)
31 \( 1 - 3.11T + 31T^{2} \)
37 \( 1 + 0.992T + 37T^{2} \)
41 \( 1 + 0.416T + 41T^{2} \)
43 \( 1 + 7.21T + 43T^{2} \)
47 \( 1 - 2.24T + 47T^{2} \)
53 \( 1 + 0.260T + 53T^{2} \)
59 \( 1 - 5.18T + 59T^{2} \)
61 \( 1 - 0.768T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 2.00T + 71T^{2} \)
73 \( 1 - 4.56T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 - 7.66T + 89T^{2} \)
97 \( 1 - 7.22T + 97T^{2} \)
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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

−8.914527657767499071949691136427, −8.035361077189361306505535396398, −7.37387608373446899439561469089, −6.57721927611342834778110626054, −5.37968145781877744742242006664, −4.92540802154531447653526525912, −4.49638397639581646690593461608, −2.27878905050750954219254568833, −1.18800623888279935406821382890, 0, 1.18800623888279935406821382890, 2.27878905050750954219254568833, 4.49638397639581646690593461608, 4.92540802154531447653526525912, 5.37968145781877744742242006664, 6.57721927611342834778110626054, 7.37387608373446899439561469089, 8.035361077189361306505535396398, 8.914527657767499071949691136427

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