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Properties

Label 2-2523-87.80-c0-0-3
Degree $2$
Conductor $2523$
Sign $0.816 + 0.577i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.816 + 0.577i$
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: \((2,\ 2523,\ (\ :0),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145542786\)
\(L(\frac12)\) \(\approx\) \(1.145542786\)
\(L(1)\) 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

−9.218278967967678883869697808399, −8.335748486746748755914117474707, −7.34454685301721989199556781571, −6.54399052222372118006773460093, −5.85729793888005566789182911659, −5.32738637724755313157521604194, −4.15800088614387320604140877175, −3.34220847010021300115302694318, −2.47036986056455500924864118256, −0.797817780164104121578615914482, 1.55985506045017868383519535106, 2.23920681372071010424298958966, 3.38196743786940801088323937550, 4.26503958304585254053289367423, 5.71138572282976293052795614068, 6.35386885357157183836253055418, 6.57524465588171530155956331555, 7.70583127243682566074662677083, 8.117085351024557606930016739052, 9.015934185907422378292819824371

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