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Properties

Label 2-1725-5.4-c1-0-28
Degree $2$
Conductor $1725$
Sign $0.447 - 0.894i$
Analytic cond. $13.7741$
Root an. cond. $3.71136$
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  + 2.44i·2-s − i·3-s − 3.99·4-s + 2.44·6-s − i·7-s − 4.89i·8-s − 9-s − 2.44·11-s + 3.99i·12-s + 0.449i·13-s + 2.44·14-s + 3.99·16-s − 0.550i·17-s − 2.44i·18-s + 0.449·19-s + ⋯
L(s)  = 1  + 1.73i·2-s − 0.577i·3-s − 1.99·4-s + 0.999·6-s − 0.377i·7-s − 1.73i·8-s − 0.333·9-s − 0.738·11-s + 1.15i·12-s + 0.124i·13-s + 0.654·14-s + 0.999·16-s − 0.133i·17-s − 0.577i·18-s + 0.103·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1725\)    =    \(3 \cdot 5^{2} \cdot 23\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(13.7741\)
Root analytic conductor: \(3.71136\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1725} (1174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1725,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.294292883\)
\(L(\frac12)\) \(\approx\) \(1.294292883\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
23 \( 1 - iT \)
good2 \( 1 - 2.44iT - 2T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
13 \( 1 - 0.449iT - 13T^{2} \)
17 \( 1 + 0.550iT - 17T^{2} \)
19 \( 1 - 0.449T + 19T^{2} \)
29 \( 1 - 4.34T + 29T^{2} \)
31 \( 1 - 9.89T + 31T^{2} \)
37 \( 1 + 5.89iT - 37T^{2} \)
41 \( 1 - 0.550T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 3.55iT - 47T^{2} \)
53 \( 1 + 5.44iT - 53T^{2} \)
59 \( 1 - 4.34T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 9.34iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 9.24iT - 83T^{2} \)
89 \( 1 - 7.10T + 89T^{2} \)
97 \( 1 - 12.8iT - 97T^{2} \)
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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

−9.127826420357734136287524675909, −8.190851377303293383980977775260, −7.892407090945584823458779362806, −6.96025220338363280833008977751, −6.51552072912482311320149074495, −5.56699250990239603258626686453, −4.91555292122897098446396119103, −3.91150379195668501090409993150, −2.49347553916233746902650245593, −0.67452413381931999040164368652, 0.913500051188016091040801826839, 2.38855946280244887698763722718, 2.93601270475772718809454508336, 3.97675454886921568300778926111, 4.75796398584492861664010531251, 5.50965510387386144050262027135, 6.70185144568402145919445791253, 8.157936916373713754287337607961, 8.593999266807196526174576407140, 9.566154755221709658197628226564

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