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Properties

Label 2-2523-87.38-c0-0-4
Degree $2$
Conductor $2523$
Sign $0.665 + 0.746i$
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

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.665 + 0.746i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1952, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ 0.665 + 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5122201512\)
\(L(\frac12)\) \(\approx\) \(0.5122201512\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.781 - 0.623i)T \)
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 \( 1 + (1.26 + 1.00i)T + (0.222 + 0.974i)T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.268 + 0.556i)T + (-0.623 + 0.781i)T^{2} \)
37 \( 1 + (1.57 + 0.360i)T + (0.900 + 0.433i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.268 + 0.556i)T + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.483 - 0.385i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (0.360 - 1.57i)T + (-0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.702 + 1.45i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 + (-1.57 - 0.360i)T + (0.900 + 0.433i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.623 - 0.781i)T^{2} \)
97 \( 1 + (0.483 + 0.385i)T + (0.222 + 0.974i)T^{2} \)
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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

−8.826436852942177220946034725667, −8.486406412072168717162943167443, −7.50433651786523532253949593636, −6.66925976798923386341446372250, −5.62310406847668846002907494416, −5.03766250440289686718638711257, −4.33933181142480179248971233736, −3.41936294008549782707666016310, −2.46179987083228635725234551543, −0.40070512339981147031590914024, 1.35371139107904117761525929787, 2.00502542637809302438337776634, 3.84757269710187825858317071556, 4.66636216698534946906930908409, 5.19953645572270958584326050078, 6.24617011483319320547059023182, 6.69828924972731566144072814460, 7.55614969226827977249914884054, 8.472071823738583754651608570286, 9.184212216027792491199427859395

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