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Properties

Label 2-525-105.89-c1-0-26
Degree $2$
Conductor $525$
Sign $0.602 + 0.797i$
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)2-s + (−0.866 + 1.5i)3-s + (−0.5 − 0.866i)4-s + (1.5 + 2.59i)6-s + (0.866 − 2.5i)7-s + 1.73·8-s + (−1.5 − 2.59i)9-s + (−3 + 1.73i)11-s + 1.73·12-s + 3.46·13-s + (−3 − 3.46i)14-s + (2.49 − 4.33i)16-s + (5.19 − 3i)17-s − 5.19·18-s + (6 + 3.46i)19-s + ⋯
L(s)  = 1  + (0.612 − 1.06i)2-s + (−0.499 + 0.866i)3-s + (−0.250 − 0.433i)4-s + (0.612 + 1.06i)6-s + (0.327 − 0.944i)7-s + 0.612·8-s + (−0.5 − 0.866i)9-s + (−0.904 + 0.522i)11-s + 0.500·12-s + 0.960·13-s + (−0.801 − 0.925i)14-s + (0.624 − 1.08i)16-s + (1.26 − 0.727i)17-s − 1.22·18-s + (1.37 + 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.602 + 0.797i$
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: \((2,\ 525,\ (\ :1/2),\ 0.602 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69509 - 0.843923i\)
\(L(\frac12)\) \(\approx\) \(1.69509 - 0.843923i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 2.5i)T \)
good2 \( 1 + (-0.866 + 1.5i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + (-5.19 + 3i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.866 + 1.5i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.73iT - 29T^{2} \)
31 \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 + 2i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
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 \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.2 - 6.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 + (1.73 + 3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−10.62213551546610694661568055542, −10.35417081870895609277313213492, −9.457790612896602963018724382350, −7.988662752998097272490630396301, −7.15624459018367723618390762043, −5.53359212035199242984853930380, −4.88554637999535604849158042519, −3.79571754599501872709566497319, −3.12520920391336131066966929690, −1.23537609397020519097240725519, 1.48803078637278665572929698782, 3.15433494293506180301455294700, 4.95853586100481030807384072809, 5.66502639374113202375159648487, 6.09749521862435195714747803608, 7.33861703284010859177609656936, 7.933299186870908890610461164933, 8.768520279000529072432469924624, 10.29380233699445215075544206955, 11.19609414413950554210188563135

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