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Properties

Label 2-525-105.89-c1-0-27
Degree $2$
Conductor $525$
Sign $0.999 + 0.0351i$
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  + (0.442 − 0.766i)2-s + (1.72 − 0.189i)3-s + (0.608 + 1.05i)4-s + (0.616 − 1.40i)6-s + (−0.206 + 2.63i)7-s + 2.84·8-s + (2.92 − 0.652i)9-s + (1.25 − 0.723i)11-s + (1.24 + 1.69i)12-s − 4.04·13-s + (1.93 + 1.32i)14-s + (0.0422 − 0.0730i)16-s + (−4.98 + 2.87i)17-s + (0.795 − 2.53i)18-s + (0.356 + 0.206i)19-s + ⋯
L(s)  = 1  + (0.312 − 0.541i)2-s + (0.993 − 0.109i)3-s + (0.304 + 0.527i)4-s + (0.251 − 0.572i)6-s + (−0.0778 + 0.996i)7-s + 1.00·8-s + (0.976 − 0.217i)9-s + (0.377 − 0.218i)11-s + (0.360 + 0.490i)12-s − 1.12·13-s + (0.515 + 0.354i)14-s + (0.0105 − 0.0182i)16-s + (−1.20 + 0.698i)17-s + (0.187 − 0.596i)18-s + (0.0818 + 0.0472i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.999 + 0.0351i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.999 + 0.0351i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55568 - 0.0449070i\)
\(L(\frac12)\) \(\approx\) \(2.55568 - 0.0449070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.189i)T \)
5 \( 1 \)
7 \( 1 + (0.206 - 2.63i)T \)
good2 \( 1 + (-0.442 + 0.766i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-1.25 + 0.723i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.04T + 13T^{2} \)
17 \( 1 + (4.98 - 2.87i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.356 - 0.206i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.65 + 6.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.82iT - 29T^{2} \)
31 \( 1 + (-2.30 + 1.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.06 + 2.92i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.46T + 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + (9.41 + 5.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.697 - 1.20i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.583 - 1.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.58 - 2.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.99 - 3.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 + (-1.57 - 2.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.42 - 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.5iT - 83T^{2} \)
89 \( 1 + (-3.90 + 6.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.86T + 97T^{2} \)
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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

−10.93455286185072438757156040720, −9.979803467183212566726931470526, −8.884831634464276990262925975794, −8.408587048075260566418445063455, −7.28915155409370963907483709985, −6.48231792083950685542873237424, −4.85208992802202563305765007627, −3.86610351348121154679718731582, −2.68613467371197739422057993540, −2.07675847064030527735697107572, 1.52481937781830611498992588007, 2.94862632529953262298234976978, 4.36510878738689705029762906440, 4.99237858848595815119030521908, 6.62443618716684888528025798160, 7.17405634427908867019462199446, 7.87685332508434454227036167048, 9.279811466948053922002769644852, 9.803562242715302246762332969374, 10.72266252033722602679250306729

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