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Properties

Label 2-525-35.9-c1-0-21
Degree $2$
Conductor $525$
Sign $0.859 + 0.510i$
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  + (1.22 + 0.707i)2-s + (0.866 − 0.5i)3-s + 1.41·6-s + (0.358 − 2.62i)7-s − 2.82i·8-s + (0.499 − 0.866i)9-s + (−0.292 − 0.507i)11-s + 4.41i·13-s + (2.29 − 2.95i)14-s + (2.00 − 3.46i)16-s + (1.94 − 1.12i)17-s + (1.22 − 0.707i)18-s + (2.32 − 4.03i)19-s + (−1 − 2.44i)21-s − 0.828i·22-s + (1.94 + 1.12i)23-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.499 − 0.288i)3-s + 0.577·6-s + (0.135 − 0.990i)7-s − 0.999i·8-s + (0.166 − 0.288i)9-s + (−0.0883 − 0.152i)11-s + 1.22i·13-s + (0.612 − 0.790i)14-s + (0.500 − 0.866i)16-s + (0.471 − 0.271i)17-s + (0.288 − 0.166i)18-s + (0.534 − 0.925i)19-s + (−0.218 − 0.534i)21-s − 0.176i·22-s + (0.404 + 0.233i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.859 + 0.510i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.859 + 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41206 - 0.662724i\)
\(L(\frac12)\) \(\approx\) \(2.41206 - 0.662724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.358 + 2.62i)T \)
good2 \( 1 + (-1.22 - 0.707i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (0.292 + 0.507i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.41iT - 13T^{2} \)
17 \( 1 + (-1.94 + 1.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.32 + 4.03i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.94 - 1.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 + (-2.91 - 5.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.28 - 4.20i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 - 7.58iT - 43T^{2} \)
47 \( 1 + (-11.5 - 6.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.91 - 3.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.24 + 3.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.8 - 6.86i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.585T + 71T^{2} \)
73 \( 1 + (-10.4 + 6.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.32 + 5.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.58iT - 83T^{2} \)
89 \( 1 + (6.12 - 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.17iT - 97T^{2} \)
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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

−10.86427550537178409326446236643, −9.708157960056449794046041070878, −9.137980847792108157645447453807, −7.72260510387443464515067321538, −7.06911027767869535403794134086, −6.26813078812495887542433081950, −4.98829982010513707696782514227, −4.20481539430060432195881509317, −3.13361779092236880977816560795, −1.23231833059893979769172051716, 2.13441258147703182257372319329, 3.12714220170478014819261771676, 4.00486449374366995632166463839, 5.30495316966036236973677216642, 5.76812450797055278484767731675, 7.61523515150538986584232966096, 8.244625499774629683486191484919, 9.161240840850786389262352765533, 10.14853094288797217000785623614, 11.09483051895559768540754591482

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