↑

Properties

Label 2-2523-87.74-c0-0-4
Degree $2$
Conductor $2523$
Sign $0.541 - 0.840i$
Analytic cond. $1.25914$
Root an. cond. $1.12211$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 + 0.781i)3-s + (0.623 + 0.781i)4-s + (0.385 − 0.483i)7-s + (−0.222 − 0.974i)9-s − 0.999·12-s + (0.360 − 1.57i)13-s + (−0.222 + 0.974i)16-s + (1.00 + 1.26i)19-s + (0.137 + 0.602i)21-s + (0.623 + 0.781i)25-s + (0.900 + 0.433i)27-s + 0.618·28-s + (0.556 + 0.268i)31-s + (0.623 − 0.781i)36-s + (−0.360 − 1.57i)37-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)3-s + (0.623 + 0.781i)4-s + (0.385 − 0.483i)7-s + (−0.222 − 0.974i)9-s − 0.999·12-s + (0.360 − 1.57i)13-s + (−0.222 + 0.974i)16-s + (1.00 + 1.26i)19-s + (0.137 + 0.602i)21-s + (0.623 + 0.781i)25-s + (0.900 + 0.433i)27-s + 0.618·28-s + (0.556 + 0.268i)31-s + (0.623 − 0.781i)36-s + (−0.360 − 1.57i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.541 - 0.840i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ 0.541 - 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.236227216\)
\(L(\frac12)\) \(\approx\) \(1.236227216\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 \)
good2 \( 1 + (-0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.623 - 0.781i)T^{2} \)
7 \( 1 + (-0.385 + 0.483i)T + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.360 + 1.57i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-1.00 - 1.26i)T + (-0.222 + 0.974i)T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.556 - 0.268i)T + (0.623 + 0.781i)T^{2} \)
37 \( 1 + (0.360 + 1.57i)T + (-0.900 + 0.433i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-0.556 + 0.268i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.385 - 0.483i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + (-0.360 - 1.57i)T + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.45 - 0.702i)T + (0.623 - 0.781i)T^{2} \)
79 \( 1 + (0.360 + 1.57i)T + (-0.900 + 0.433i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + (0.385 + 0.483i)T + (-0.222 + 0.974i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.207666066253274177481237252791, −8.385842381976723793781438147497, −7.63533128640205098894508634954, −7.05817321738763809236514701885, −5.87760551059066289943782186325, −5.49026392587385315607642269217, −4.32839255489256005902763455652, −3.55346669532149213291090627043, −2.90444615070095719423734966900, −1.23382384990329396002934630180, 1.10211455181904353001206408030, 2.00624993538313942776420144116, 2.85628133501854145740084875453, 4.59194750854515835931590440458, 5.10429296470493059416502063353, 6.06228639987318004711834698861, 6.65394948847634768803163963847, 7.13653646784660417506425888378, 8.130085008014180474764815858955, 8.979469346418507528509491267732

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