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Properties

Label 2-1805-5.4-c1-0-72
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

\(L(1)\) \(\approx\) \(2.621796978\)
\(L(\frac12)\) \(\approx\) \(2.621796978\)
\(L(\frac{3}{2})\) 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

−9.391834011887580513848135774649, −8.950758007797051526505075302740, −7.74116791775195077704265855020, −7.04949026532880443317325580171, −6.51952618699298928813546875080, −5.52428568759313228466654693757, −5.14382204489898012483845864506, −3.66745385958929722713050697489, −2.55448244410996841148406212285, −1.81043276448645367723951790668, 1.04478485693328233626836055214, 1.57731315511407829231185876902, 2.73351192642314388735976543719, 4.00703716007821906348018997510, 4.51465559558716460050515107411, 5.94474759670010324891290690539, 6.55901090545799221635771316340, 7.35173224551082936657579776195, 8.143727453942252105989494673514, 9.225234320820718063485656878994

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