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Properties

Label 2-525-35.4-c1-0-19
Degree $2$
Conductor $525$
Sign $-0.758 + 0.652i$
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  + (0.633 − 0.366i)2-s + (−0.866 − 0.5i)3-s + (−0.732 + 1.26i)4-s − 0.732·6-s + (−2.5 − 0.866i)7-s + 2.53i·8-s + (0.499 + 0.866i)9-s + (1.36 − 2.36i)11-s + (1.26 − 0.732i)12-s − 5.73i·13-s + (−1.90 + 0.366i)14-s + (−0.535 − 0.928i)16-s + (−5.83 − 3.36i)17-s + (0.633 + 0.366i)18-s + (−1.23 − 2.13i)19-s + ⋯
L(s)  = 1  + (0.448 − 0.258i)2-s + (−0.499 − 0.288i)3-s + (−0.366 + 0.633i)4-s − 0.298·6-s + (−0.944 − 0.327i)7-s + 0.896i·8-s + (0.166 + 0.288i)9-s + (0.411 − 0.713i)11-s + (0.366 − 0.211i)12-s − 1.58i·13-s + (−0.508 + 0.0978i)14-s + (−0.133 − 0.232i)16-s + (−1.41 − 0.816i)17-s + (0.149 + 0.0862i)18-s + (−0.282 − 0.489i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.758 + 0.652i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.758 + 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.215755 - 0.581744i\)
\(L(\frac12)\) \(\approx\) \(0.215755 - 0.581744i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
good2 \( 1 + (-0.633 + 0.366i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-1.36 + 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.73iT - 13T^{2} \)
17 \( 1 + (5.83 + 3.36i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.23 + 2.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.09 + 0.633i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.19T + 29T^{2} \)
31 \( 1 + (3.23 - 5.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.23 + 3.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.73T + 41T^{2} \)
43 \( 1 - 7.19iT - 43T^{2} \)
47 \( 1 + (-1.73 + i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.26 + 4.19i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.09 + 8.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.30 + 1.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.19T + 71T^{2} \)
73 \( 1 + (4.03 + 2.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.69 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.12iT - 83T^{2} \)
89 \( 1 + (4.56 + 7.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.07iT - 97T^{2} \)
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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

−10.96062466545782672145487382356, −9.595292848255758697790981369123, −8.773440103199969200876033646494, −7.73604954991511620515986225357, −6.81042467943676085693815140863, −5.79357867819380218714671732897, −4.77102180321959370430531603636, −3.60187576032734780468868869338, −2.68983911703165740175351440803, −0.32527822570076160909663539011, 1.91278886851443463868453784016, 3.97774006766146059012626886139, 4.42894267632393842110637277179, 5.80443953973680155959918655087, 6.38997482610495799174715210156, 7.15716781910576466565254516829, 9.042900087548668521446206369498, 9.320031417367857641298407060510, 10.25546845465116237180724521265, 11.18290753853724786671205718892

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