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Properties

Label 2-2523-87.5-c0-0-5
Degree $2$
Conductor $2523$
Sign $-0.998 + 0.0495i$
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  + (−0.974 + 0.222i)3-s + (0.222 − 0.974i)4-s + (−0.137 − 0.602i)7-s + (0.900 − 0.433i)9-s + i·12-s + (−1.45 − 0.702i)13-s + (−0.900 − 0.433i)16-s + (−1.57 − 0.360i)19-s + (0.268 + 0.556i)21-s + (−0.222 + 0.974i)25-s + (−0.781 + 0.623i)27-s − 0.618·28-s + (0.483 − 0.385i)31-s + (−0.222 − 0.974i)36-s + (0.702 + 1.45i)37-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)3-s + (0.222 − 0.974i)4-s + (−0.137 − 0.602i)7-s + (0.900 − 0.433i)9-s + i·12-s + (−1.45 − 0.702i)13-s + (−0.900 − 0.433i)16-s + (−1.57 − 0.360i)19-s + (0.268 + 0.556i)21-s + (−0.222 + 0.974i)25-s + (−0.781 + 0.623i)27-s − 0.618·28-s + (0.483 − 0.385i)31-s + (−0.222 − 0.974i)36-s + (0.702 + 1.45i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $-0.998 + 0.0495i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1745, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ -0.998 + 0.0495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3359382533\)
\(L(\frac12)\) \(\approx\) \(0.3359382533\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.974 - 0.222i)T \)
29 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
7 \( 1 + (0.137 + 0.602i)T + (-0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (1.45 + 0.702i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (1.57 + 0.360i)T + (0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.483 + 0.385i)T + (0.222 - 0.974i)T^{2} \)
37 \( 1 + (-0.702 - 1.45i)T + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.483 + 0.385i)T + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.602 - 0.137i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (1.45 - 0.702i)T + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (1.26 + 1.00i)T + (0.222 + 0.974i)T^{2} \)
79 \( 1 + (0.702 + 1.45i)T + (-0.623 + 0.781i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.222 + 0.974i)T^{2} \)
97 \( 1 + (0.602 + 0.137i)T + (0.900 + 0.433i)T^{2} \)
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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

−8.953017645285392542304450402164, −7.69057438958405588775035681286, −7.01813902484060018116721467357, −6.34034978396148806400783930557, −5.61637063204009888949230578463, −4.80584813226122737149054685751, −4.27778259241786562185171337656, −2.82134035216879979010721181071, −1.58068628289035731063943644014, −0.23579361826986376441570693428, 1.99938036888486948333436947948, 2.66774093728438258676815737183, 4.16991639419571531707540813933, 4.57673440509508641815459821595, 5.72890020287675841636861433267, 6.46772576387370916003784096574, 7.09680113185899957424189483444, 7.82966673906420623055478213135, 8.617078210803652856893202883438, 9.455703570988303826982099973474

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