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Properties

Label 2-525-35.19-c2-0-24
Degree $2$
Conductor $525$
Sign $-0.742 - 0.669i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.98 + 1.72i)2-s + (0.866 + 1.5i)3-s + (3.94 + 6.84i)4-s + 5.97i·6-s + (2.59 + 6.5i)7-s + 13.4i·8-s + (−1.5 + 2.59i)9-s + (−1.44 − 2.51i)11-s + (−6.84 + 11.8i)12-s − 17.1·13-s + (−3.44 + 23.8i)14-s + (−7.39 + 12.8i)16-s + (−0.953 − 1.65i)17-s + (−8.96 + 5.17i)18-s + (14.5 + 8.39i)19-s + ⋯
L(s)  = 1  + (1.49 + 0.862i)2-s + (0.288 + 0.5i)3-s + (0.987 + 1.71i)4-s + 0.995i·6-s + (0.371 + 0.928i)7-s + 1.68i·8-s + (−0.166 + 0.288i)9-s + (−0.131 − 0.228i)11-s + (−0.570 + 0.987i)12-s − 1.31·13-s + (−0.246 + 1.70i)14-s + (−0.462 + 0.800i)16-s + (−0.0560 − 0.0971i)17-s + (−0.497 + 0.287i)18-s + (0.765 + 0.441i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.742 - 0.669i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.742 - 0.669i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.399182434\)
\(L(\frac12)\) \(\approx\) \(4.399182434\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.59 - 6.5i)T \)
good2 \( 1 + (-2.98 - 1.72i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (1.44 + 2.51i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 17.1T + 169T^{2} \)
17 \( 1 + (0.953 + 1.65i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.5 - 8.39i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-17.3 - 10i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 31.3T + 841T^{2} \)
31 \( 1 + (-29.3 + 16.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (42.8 + 24.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 76.7iT - 1.68e3T^{2} \)
43 \( 1 - 59.7iT - 1.84e3T^{2} \)
47 \( 1 + (33.5 - 58.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-82.6 + 47.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (60.1 - 34.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (36.4 + 21.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-81.4 + 47.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 22.9T + 5.04e3T^{2} \)
73 \( 1 + (21.4 + 37.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-10.0 + 17.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 0.857T + 6.88e3T^{2} \)
89 \( 1 + (18.7 + 10.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 72.3T + 9.40e3T^{2} \)
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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

−11.37342284346689136394583497358, −10.06789657501083914521892363617, −9.067203304712713492491904052741, −8.012570301649549785332057275313, −7.25252708981133039627999321359, −6.09450068340932494680644890086, −5.19313397736493999982891514905, −4.68839885461675020179588890534, −3.36815909262531305041610549305, −2.47040470990819259017132722462, 1.09445418548198148657173364678, 2.42127747947528427620690313813, 3.33428013596492342248772488966, 4.61657188873912770738403840937, 5.11316690054074772851314801343, 6.60193259074046643836415143547, 7.25516520750185194622405342097, 8.449897036770032626112450195265, 9.934766231875628822036944090397, 10.46751121904385259600874235162

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