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Properties

Label 2-525-105.32-c1-0-33
Degree $2$
Conductor $525$
Sign $-0.426 + 0.904i$
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  + (−2.35 − 0.631i)2-s + (1.54 − 0.775i)3-s + (3.42 + 1.97i)4-s + (−4.13 + 0.849i)6-s + (1.91 − 1.82i)7-s + (−3.36 − 3.36i)8-s + (1.79 − 2.40i)9-s + (−3.08 − 1.77i)11-s + (6.83 + 0.406i)12-s + (−1.28 + 1.28i)13-s + (−5.67 + 3.08i)14-s + (1.85 + 3.21i)16-s + (−0.792 − 2.95i)17-s + (−5.75 + 4.52i)18-s + (0.331 − 0.191i)19-s + ⋯
L(s)  = 1  + (−1.66 − 0.446i)2-s + (0.894 − 0.447i)3-s + (1.71 + 0.987i)4-s + (−1.68 + 0.346i)6-s + (0.725 − 0.688i)7-s + (−1.19 − 1.19i)8-s + (0.599 − 0.800i)9-s + (−0.928 − 0.536i)11-s + (1.97 + 0.117i)12-s + (−0.356 + 0.356i)13-s + (−1.51 + 0.823i)14-s + (0.463 + 0.803i)16-s + (−0.192 − 0.717i)17-s + (−1.35 + 1.06i)18-s + (0.0761 − 0.0439i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.426 + 0.904i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.426 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.454983 - 0.717563i\)
\(L(\frac12)\) \(\approx\) \(0.454983 - 0.717563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.54 + 0.775i)T \)
5 \( 1 \)
7 \( 1 + (-1.91 + 1.82i)T \)
good2 \( 1 + (2.35 + 0.631i)T + (1.73 + i)T^{2} \)
11 \( 1 + (3.08 + 1.77i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.28 - 1.28i)T - 13iT^{2} \)
17 \( 1 + (0.792 + 2.95i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.331 + 0.191i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.658 + 2.45i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.51T + 29T^{2} \)
31 \( 1 + (-0.323 + 0.561i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.34 + 5.00i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (-0.335 + 0.335i)T - 43iT^{2} \)
47 \( 1 + (2.80 + 0.751i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.04 + 0.815i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.81 + 6.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.45 + 9.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.3 - 3.31i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.06iT - 71T^{2} \)
73 \( 1 + (-0.849 - 3.17i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.21 - 1.85i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.973 - 0.973i)T + 83iT^{2} \)
89 \( 1 + (-1.51 - 2.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.3 - 10.3i)T + 97iT^{2} \)
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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

−10.36840478366603478637397848295, −9.622389135019257372325424581849, −8.729853538738128235069675160254, −8.039093506169542786578755609003, −7.48077373468627034976785473193, −6.60247259110152183501889362352, −4.69736162312615691591435310681, −3.08350223428749731434773069313, −2.11821973974800748685113959421, −0.78185260655787997983374329136, 1.73914002376074862184833152357, 2.75575311934874522123456836811, 4.60950923747197738846478569319, 5.75762447488437032878779130023, 7.20228758962265055645400019454, 7.84584983113035563048593191473, 8.542167681261378540072115255108, 9.130474406087089829820676151383, 10.22760512689162687324219696993, 10.46016648065630938621692700571

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