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Properties

Label 2-525-21.5-c1-0-33
Degree $2$
Conductor $525$
Sign $0.716 - 0.697i$
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  + (2.36 + 1.36i)2-s + (0.310 − 1.70i)3-s + (2.73 + 4.74i)4-s + (3.06 − 3.61i)6-s + (2.24 − 1.39i)7-s + 9.49i·8-s + (−2.80 − 1.05i)9-s + (−1.79 + 1.03i)11-s + (8.92 − 3.19i)12-s − 2.04i·13-s + (7.22 − 0.226i)14-s + (−7.50 + 13.0i)16-s + (1.05 + 1.82i)17-s + (−5.20 − 6.34i)18-s + (−2.41 − 1.39i)19-s + ⋯
L(s)  = 1  + (1.67 + 0.966i)2-s + (0.179 − 0.983i)3-s + (1.36 + 2.37i)4-s + (1.25 − 1.47i)6-s + (0.849 − 0.526i)7-s + 3.35i·8-s + (−0.935 − 0.352i)9-s + (−0.540 + 0.312i)11-s + (2.57 − 0.921i)12-s − 0.566i·13-s + (1.93 − 0.0604i)14-s + (−1.87 + 3.25i)16-s + (0.255 + 0.441i)17-s + (−1.22 − 1.49i)18-s + (−0.553 − 0.319i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.716 - 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.55338 + 1.44334i\)
\(L(\frac12)\) \(\approx\) \(3.55338 + 1.44334i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.310 + 1.70i)T \)
5 \( 1 \)
7 \( 1 + (-2.24 + 1.39i)T \)
good2 \( 1 + (-2.36 - 1.36i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (1.79 - 1.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.04iT - 13T^{2} \)
17 \( 1 + (-1.05 - 1.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.41 + 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.53 + 0.888i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.79iT - 29T^{2} \)
31 \( 1 + (-6.04 + 3.49i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.03 - 5.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 + 4.37T + 43T^{2} \)
47 \( 1 + (1.47 - 2.55i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.36 + 3.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.44 - 2.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.30 + 3.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.70 - 6.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.80iT - 71T^{2} \)
73 \( 1 + (-5.81 + 3.35i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.95 + 5.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.96T + 83T^{2} \)
89 \( 1 + (-6.20 + 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.97iT - 97T^{2} \)
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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

−11.54378868010949517457722423958, −10.41631252456730898326790514844, −8.298120396956597978453963793560, −8.080111509901924472909725468388, −7.11715213823882659182696129783, −6.37852684137895251324774181239, −5.39457812669674021709876297898, −4.52843132347181699192635963112, −3.30342853005241220876945889019, −2.10762436456530077271080961216, 1.94011229130507800257371489024, 3.01170305270152387096738168768, 4.00794332603727963072538615331, 4.99036276388097807892201571957, 5.42772711274311914623426033428, 6.59954875225441742123341596388, 8.223830214983302821967361201709, 9.341375495650962370776439978819, 10.37717419446824594660960258136, 10.83887421466201190232845845844

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