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Properties

Label 2-525-25.16-c1-0-7
Degree $2$
Conductor $525$
Sign $0.929 + 0.369i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.66 − 1.20i)2-s + (0.309 + 0.951i)3-s + (0.687 + 2.11i)4-s + (2.21 + 0.282i)5-s + (0.635 − 1.95i)6-s + 7-s + (0.141 − 0.436i)8-s + (−0.809 + 0.587i)9-s + (−3.34 − 3.14i)10-s + (−1.63 − 1.18i)11-s + (−1.79 + 1.30i)12-s + (4.82 − 3.50i)13-s + (−1.66 − 1.20i)14-s + (0.417 + 2.19i)15-s + (2.83 − 2.05i)16-s + (−0.186 + 0.572i)17-s + ⋯
L(s)  = 1  + (−1.17 − 0.854i)2-s + (0.178 + 0.549i)3-s + (0.343 + 1.05i)4-s + (0.992 + 0.126i)5-s + (0.259 − 0.797i)6-s + 0.377·7-s + (0.0501 − 0.154i)8-s + (−0.269 + 0.195i)9-s + (−1.05 − 0.995i)10-s + (−0.493 − 0.358i)11-s + (−0.519 + 0.377i)12-s + (1.33 − 0.972i)13-s + (−0.444 − 0.322i)14-s + (0.107 + 0.567i)15-s + (0.708 − 0.514i)16-s + (−0.0451 + 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.929 + 0.369i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01930 - 0.195390i\)
\(L(\frac12)\) \(\approx\) \(1.01930 - 0.195390i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-2.21 - 0.282i)T \)
7 \( 1 - T \)
good2 \( 1 + (1.66 + 1.20i)T + (0.618 + 1.90i)T^{2} \)
11 \( 1 + (1.63 + 1.18i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-4.82 + 3.50i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.186 - 0.572i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.44 - 4.43i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.298 + 0.216i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.78 - 5.48i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.18 - 6.72i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.38 + 6.08i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.90 + 1.38i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + (2.90 + 8.92i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.449 - 1.38i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.23 + 0.898i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.65 + 1.92i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.82 - 8.70i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.33 + 7.18i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.44 + 1.04i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.352 + 1.08i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.538 - 1.65i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-12.8 - 9.34i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.40 + 4.32i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.69893322518044778774179631251, −10.11776845344522041463751657779, −9.062680083530372991984508236759, −8.565339209992441290810590836036, −7.68738151571260015313399490646, −6.03401083039224482406259405640, −5.31134516056267269342067785455, −3.59134229189468132129865435887, −2.52216234217458072558113951173, −1.25307445417502299184677265406, 1.13163056284052688161959907749, 2.42261495012236236881240133498, 4.40921822146770201804553754486, 5.95734579444609077696509096378, 6.40146265453169087192841412573, 7.45545794412488870769194434032, 8.227555259335102937713830515811, 9.115918053443692628911101839207, 9.558971281763548139343249804728, 10.69855095057409306325596181269

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