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Properties

Label 2-525-105.59-c1-0-36
Degree $2$
Conductor $525$
Sign $-0.621 + 0.783i$
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  + (−0.866 + 1.5i)3-s + (1 − 1.73i)4-s + (−1.73 − 2i)7-s + (−1.5 − 2.59i)9-s + (1.73 + 3i)12-s − 6.92·13-s + (−1.99 − 3.46i)16-s + (−3 + 1.73i)19-s + (4.5 − 0.866i)21-s + 5.19·27-s + (−5.19 + 0.999i)28-s + (−7.5 − 4.33i)31-s − 6·36-s + (9.52 − 5.5i)37-s + (5.99 − 10.3i)39-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.654 − 0.755i)7-s + (−0.5 − 0.866i)9-s + (0.499 + 0.866i)12-s − 1.92·13-s + (−0.499 − 0.866i)16-s + (−0.688 + 0.397i)19-s + (0.981 − 0.188i)21-s + 1.00·27-s + (−0.981 + 0.188i)28-s + (−1.34 − 0.777i)31-s − 36-s + (1.56 − 0.904i)37-s + (0.960 − 1.66i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.621 + 0.783i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.235801 - 0.487881i\)
\(L(\frac12)\) \(\approx\) \(0.235801 - 0.487881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 1.5i)T \)
5 \( 1 \)
7 \( 1 + (1.73 + 2i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.92T + 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (7.5 + 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.52 + 5.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 5iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-13.5 + 7.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.8 + 8i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-0.866 + 1.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.5 - 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.19T + 97T^{2} \)
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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

−10.49929618764392977475927995598, −9.755990565740178574145286843845, −9.326224027279164045446005544686, −7.60567950771521789978397705076, −6.75151195981889867314297622062, −5.83241277213078958917455451227, −4.92203804438125853298384469613, −3.90193060525851594972408428590, −2.42035122585773161268303658130, −0.30359770373932817494901780829, 2.18997881710794867244056078055, 2.94234495738880880291176160224, 4.66220408144396368763745782097, 5.82439314737636279219987513629, 6.79708566431453852598271358562, 7.38226228244397334552178217688, 8.317199487799140816922374302173, 9.308661588376410594939309731042, 10.43289409890870346260851058160, 11.54627616559118049549194105538

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