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Properties

Label 2-525-7.2-c1-0-21
Degree $2$
Conductor $525$
Sign $-0.749 + 0.661i$
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  + (0.707 − 1.22i)2-s + (−0.5 − 0.866i)3-s − 1.41·6-s + (−1.62 − 2.09i)7-s + 2.82·8-s + (−0.499 + 0.866i)9-s + (−1.70 − 2.95i)11-s + 1.58·13-s + (−3.70 + 0.507i)14-s + (2.00 − 3.46i)16-s + (−3.12 − 5.40i)17-s + (0.707 + 1.22i)18-s + (3.32 − 5.76i)19-s + (−0.999 + 2.44i)21-s − 4.82·22-s + (−3.12 + 5.40i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)2-s + (−0.288 − 0.499i)3-s − 0.577·6-s + (−0.612 − 0.790i)7-s + 0.999·8-s + (−0.166 + 0.288i)9-s + (−0.514 − 0.891i)11-s + 0.439·13-s + (−0.990 + 0.135i)14-s + (0.500 − 0.866i)16-s + (−0.757 − 1.31i)17-s + (0.166 + 0.288i)18-s + (0.763 − 1.32i)19-s + (−0.218 + 0.534i)21-s − 1.02·22-s + (−0.650 + 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.563259 - 1.48970i\)
\(L(\frac12)\) \(\approx\) \(0.563259 - 1.48970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (1.62 + 2.09i)T \)
good2 \( 1 + (-0.707 + 1.22i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.58T + 13T^{2} \)
17 \( 1 + (3.12 + 5.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.32 + 5.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.12 - 5.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.242T + 29T^{2} \)
31 \( 1 + (-0.0857 - 0.148i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.79 - 4.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + (-4.65 + 8.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.585 - 1.01i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.24 - 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.86 - 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.41T + 71T^{2} \)
73 \( 1 + (-1.03 - 1.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.32 + 4.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.41T + 83T^{2} \)
89 \( 1 + (-1.87 + 3.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−10.90588802529745747312147923934, −9.927085331084794889391152350150, −8.827562796553232064185472195894, −7.50915357451069729783977342355, −7.05558052780846275197310253356, −5.75556878130513889407475945658, −4.62101962538669186801178569230, −3.44207219437497943693608926365, −2.56817056252267954857911553392, −0.839952529568944702235911553116, 2.08236511203292890354092003344, 3.80927029638126628561973823840, 4.76185966475079827734485526396, 5.92277023270848668495261901851, 6.17991254443848803928791093872, 7.44011287278583518695134941053, 8.392054238402755810561902833451, 9.505379659966368263317075515611, 10.35530029439501957224613322302, 10.97440616373065905100867348407

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