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Properties

Label 2-2523-87.80-c0-0-5
Degree $2$
Conductor $2523$
Sign $-0.0973 + 0.995i$
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  + (0.433 − 0.900i)3-s + (0.900 − 0.433i)4-s + (−0.556 − 0.268i)7-s + (−0.623 − 0.781i)9-s − i·12-s + (1.00 − 1.26i)13-s + (0.623 − 0.781i)16-s + (0.702 + 1.45i)19-s + (−0.483 + 0.385i)21-s + (−0.900 + 0.433i)25-s + (−0.974 + 0.222i)27-s − 0.618·28-s + (0.602 − 0.137i)31-s + (−0.900 − 0.433i)36-s + (−1.26 + 1.00i)37-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)3-s + (0.900 − 0.433i)4-s + (−0.556 − 0.268i)7-s + (−0.623 − 0.781i)9-s − i·12-s + (1.00 − 1.26i)13-s + (0.623 − 0.781i)16-s + (0.702 + 1.45i)19-s + (−0.483 + 0.385i)21-s + (−0.900 + 0.433i)25-s + (−0.974 + 0.222i)27-s − 0.618·28-s + (0.602 − 0.137i)31-s + (−0.900 − 0.433i)36-s + (−1.26 + 1.00i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0973 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0973 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $-0.0973 + 0.995i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ -0.0973 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.639449262\)
\(L(\frac12)\) \(\approx\) \(1.639449262\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.433 + 0.900i)T \)
29 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.900 - 0.433i)T^{2} \)
7 \( 1 + (0.556 + 0.268i)T + (0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (-1.00 + 1.26i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.702 - 1.45i)T + (-0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.602 + 0.137i)T + (0.900 - 0.433i)T^{2} \)
37 \( 1 + (1.26 - 1.00i)T + (0.222 - 0.974i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.602 + 0.137i)T + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.222 - 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.268 + 0.556i)T + (-0.623 - 0.781i)T^{2} \)
67 \( 1 + (-1.00 - 1.26i)T + (-0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (1.57 + 0.360i)T + (0.900 + 0.433i)T^{2} \)
79 \( 1 + (-1.26 + 1.00i)T + (0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.900 + 0.433i)T^{2} \)
97 \( 1 + (-0.268 - 0.556i)T + (-0.623 + 0.781i)T^{2} \)
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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

−8.655777893140547970382949901468, −7.983727933174301285196993307849, −7.41146052023648670673670186475, −6.54866677544466390433302975550, −5.99480334113009229821182748474, −5.34998041720959555556160070022, −3.52877138425840324903883991580, −3.22231622867700364618763521247, −1.95655758618763198208599369894, −1.06639215672873844562521664396, 1.88415597651392593574899784920, 2.81365059398152046657690665603, 3.56593304083867216466966271891, 4.32871403383070492738463209082, 5.40144430059670496196630694877, 6.32692504091318866144920850982, 6.93227676679743047129208771899, 7.85511118235086202511699000384, 8.699469295496993048962784356550, 9.199121224682139446856165088137

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