Properties

Label 2-1805-95.89-c0-0-2
Degree $2$
Conductor $1805$
Sign $0.486 + 0.873i$
Analytic cond. $0.900812$
Root an. cond. $0.949111$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.08 − 0.909i)2-s + (1.32 + 0.483i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (1.87 − 0.684i)6-s + (0.766 + 0.642i)9-s + (−1.08 − 0.909i)10-s + (0.707 − 1.22i)12-s + (−1.32 + 0.483i)13-s + (0.245 − 1.39i)15-s + (0.939 + 0.342i)16-s + 1.41·18-s − 1.00·20-s + (−0.939 + 0.342i)25-s + (−0.999 + 1.73i)26-s + ⋯
L(s)  = 1  + (1.08 − 0.909i)2-s + (1.32 + 0.483i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (1.87 − 0.684i)6-s + (0.766 + 0.642i)9-s + (−1.08 − 0.909i)10-s + (0.707 − 1.22i)12-s + (−1.32 + 0.483i)13-s + (0.245 − 1.39i)15-s + (0.939 + 0.342i)16-s + 1.41·18-s − 1.00·20-s + (−0.939 + 0.342i)25-s + (−0.999 + 1.73i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.486 + 0.873i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ 0.486 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.813726929\)
\(L(\frac12)\) \(\approx\) \(2.813726929\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 \)
good2 \( 1 + (-1.08 + 0.909i)T + (0.173 - 0.984i)T^{2} \)
3 \( 1 + (-1.32 - 0.483i)T + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.32 - 0.483i)T + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.245 - 1.39i)T + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-1.08 - 0.909i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (1.08 - 0.909i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.459205053943255757750043562125, −8.617073162850434549119662257261, −8.025293107842212973742196627214, −7.09070640146375918803291402947, −5.60040353094787667422337801153, −4.81958902432599344177310013349, −4.22872587354002815513515369891, −3.46789926067049238530870608720, −2.56457240142610405216011175310, −1.74604162140077942171069522001, 2.07527952267958783887988050643, 3.03669229030833191318267369680, 3.60026943015652998909238384991, 4.70099106110708516746914141548, 5.60350246482984774195383997464, 6.65683833250417772839819760457, 7.16364985983747647654897661927, 7.75077807100674449674393605284, 8.382511688466343796965118372962, 9.564011820627474307912052632930

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