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Properties

Label 2-1805-5.4-c1-0-88
Degree $2$
Conductor $1805$
Sign $0.738 + 0.673i$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.04i·2-s − 0.531i·3-s + 0.902·4-s + (1.65 + 1.50i)5-s − 0.556·6-s + 2.74i·7-s − 3.04i·8-s + 2.71·9-s + (1.57 − 1.73i)10-s + 0.832·11-s − 0.479i·12-s − 0.610i·13-s + 2.87·14-s + (0.800 − 0.877i)15-s − 1.37·16-s − 4.83i·17-s + ⋯
L(s)  = 1  − 0.740i·2-s − 0.306i·3-s + 0.451·4-s + (0.738 + 0.673i)5-s − 0.227·6-s + 1.03i·7-s − 1.07i·8-s + 0.905·9-s + (0.499 − 0.547i)10-s + 0.251·11-s − 0.138i·12-s − 0.169i·13-s + 0.767·14-s + (0.206 − 0.226i)15-s − 0.344·16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.738 + 0.673i$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 0.738 + 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.662801807\)
\(L(\frac12)\) \(\approx\) \(2.662801807\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.65 - 1.50i)T \)
19 \( 1 \)
good2 \( 1 + 1.04iT - 2T^{2} \)
3 \( 1 + 0.531iT - 3T^{2} \)
7 \( 1 - 2.74iT - 7T^{2} \)
11 \( 1 - 0.832T + 11T^{2} \)
13 \( 1 + 0.610iT - 13T^{2} \)
17 \( 1 + 4.83iT - 17T^{2} \)
23 \( 1 - 3.75iT - 23T^{2} \)
29 \( 1 + 3.97T + 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 - 4.33iT - 37T^{2} \)
41 \( 1 + 5.31T + 41T^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + 3.40iT - 47T^{2} \)
53 \( 1 - 13.6iT - 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 9.07T + 61T^{2} \)
67 \( 1 - 10.9iT - 67T^{2} \)
71 \( 1 - 0.677T + 71T^{2} \)
73 \( 1 - 7.01iT - 73T^{2} \)
79 \( 1 - 3.47T + 79T^{2} \)
83 \( 1 + 4.97iT - 83T^{2} \)
89 \( 1 + 5.88T + 89T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−9.548299019274129144787563634546, −8.547609049698318200413931461168, −7.24072491816323628990564610984, −6.96339725245219924978940599545, −6.00964893732055457316805820013, −5.24824456991166689469367825052, −3.88871139841380118033005184664, −2.83771737594506489248903785404, −2.23385498026220869594187732866, −1.26179392767944186165814992245, 1.21254055505406939049907491287, 2.18008115134213912779064916620, 3.75013388543775440679319229193, 4.55506399469590459068680153140, 5.35728839198478031919248770781, 6.44748720761467476898672065032, 6.72670130430569539064014174225, 7.81696737184845114401700558156, 8.368536622520933114250878280425, 9.378950467081933273058972005469

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