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Properties

Label 2-525-21.20-c1-0-8
Degree $2$
Conductor $525$
Sign $-0.544 - 0.839i$
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  + 1.09i·2-s + (−1.70 + 0.323i)3-s + 0.791·4-s + (−0.355 − 1.87i)6-s + (−1 − 2.44i)7-s + 3.06i·8-s + (2.79 − 1.09i)9-s + 3.06i·11-s + (−1.34 + 0.255i)12-s + 2.44i·13-s + (2.69 − 1.09i)14-s − 1.79·16-s + 2.69·17-s + (1.20 + 3.06i)18-s + 4.38i·19-s + ⋯
L(s)  = 1  + 0.777i·2-s + (−0.982 + 0.186i)3-s + 0.395·4-s + (−0.144 − 0.763i)6-s + (−0.377 − 0.925i)7-s + 1.08i·8-s + (0.930 − 0.366i)9-s + 0.925i·11-s + (−0.388 + 0.0737i)12-s + 0.679i·13-s + (0.719 − 0.293i)14-s − 0.447·16-s + 0.653·17-s + (0.284 + 0.723i)18-s + 1.00i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.544 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.544 - 0.839i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.544 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518241 + 0.953623i\)
\(L(\frac12)\) \(\approx\) \(0.518241 + 0.953623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.323i)T \)
5 \( 1 \)
7 \( 1 + (1 + 2.44i)T \)
good2 \( 1 - 1.09iT - 2T^{2} \)
11 \( 1 - 3.06iT - 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 - 4.38iT - 19T^{2} \)
23 \( 1 - 5.26iT - 23T^{2} \)
29 \( 1 + 5.26iT - 29T^{2} \)
31 \( 1 - 6.83iT - 31T^{2} \)
37 \( 1 - 8.58T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 6.58T + 43T^{2} \)
47 \( 1 - 2.69T + 47T^{2} \)
53 \( 1 + 3.93iT - 53T^{2} \)
59 \( 1 - 7.51T + 59T^{2} \)
61 \( 1 + 6.83iT - 61T^{2} \)
67 \( 1 - 4.16T + 67T^{2} \)
71 \( 1 - 3.06iT - 71T^{2} \)
73 \( 1 - 16.1iT - 73T^{2} \)
79 \( 1 - 0.582T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + 7.51T + 89T^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
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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

−11.25131963410150692923806949123, −10.13380769976993805946112104577, −9.758779134858978906510841496352, −8.128834182894990971101589930479, −7.21215477765514134610245761457, −6.70793807572154911134616063786, −5.76341342726780786474376875196, −4.79870501413487606067128028954, −3.66177195933941277796528526791, −1.62886288314490937644865558087, 0.73987727956380848905890537214, 2.38043841615798454625365469454, 3.43895996258404152087001431936, 4.99115655813644799878623142399, 5.99886889699928871924147088753, 6.62259548178749232333074959021, 7.79580509191869716442058631480, 8.998487298941937557159096406663, 10.05171270518536826270415832390, 10.70462238069656851516401636205

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