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Properties

Label 2-525-105.23-c1-0-39
Degree $2$
Conductor $525$
Sign $-0.986 + 0.163i$
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  + (1.46 − 0.391i)2-s + (−1.49 + 0.879i)3-s + (0.246 − 0.142i)4-s + (−1.83 + 1.86i)6-s + (−2.36 − 1.17i)7-s + (−1.83 + 1.83i)8-s + (1.45 − 2.62i)9-s + (−0.791 + 0.457i)11-s + (−0.243 + 0.429i)12-s + (−3.07 − 3.07i)13-s + (−3.92 − 0.791i)14-s + (−2.24 + 3.88i)16-s + (−0.311 + 1.16i)17-s + (1.09 − 4.40i)18-s + (−5.95 − 3.43i)19-s + ⋯
L(s)  = 1  + (1.03 − 0.276i)2-s + (−0.861 + 0.507i)3-s + (0.123 − 0.0712i)4-s + (−0.749 + 0.762i)6-s + (−0.895 − 0.444i)7-s + (−0.648 + 0.648i)8-s + (0.484 − 0.874i)9-s + (−0.238 + 0.137i)11-s + (−0.0702 + 0.124i)12-s + (−0.854 − 0.854i)13-s + (−1.04 − 0.211i)14-s + (−0.561 + 0.971i)16-s + (−0.0755 + 0.281i)17-s + (0.258 − 1.03i)18-s + (−1.36 − 0.788i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.986 + 0.163i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.986 + 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00545282 - 0.0661171i\)
\(L(\frac12)\) \(\approx\) \(0.00545282 - 0.0661171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.49 - 0.879i)T \)
5 \( 1 \)
7 \( 1 + (2.36 + 1.17i)T \)
good2 \( 1 + (-1.46 + 0.391i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.791 - 0.457i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.07 + 3.07i)T + 13iT^{2} \)
17 \( 1 + (0.311 - 1.16i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.95 + 3.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.505 - 1.88i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.72T + 29T^{2} \)
31 \( 1 + (2.31 + 4.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.207 + 0.774i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.922iT - 41T^{2} \)
43 \( 1 + (-4.80 - 4.80i)T + 43iT^{2} \)
47 \( 1 + (10.1 - 2.71i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-10.6 - 2.85i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.94 - 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.533 + 0.924i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.83 + 1.83i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 0.557iT - 71T^{2} \)
73 \( 1 + (-0.564 + 2.10i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.62 + 1.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.38 - 2.38i)T - 83iT^{2} \)
89 \( 1 + (-5.64 + 9.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.58 + 1.58i)T - 97iT^{2} \)
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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

−10.62047890273176587366290772496, −9.806814047769988634192779575421, −8.897310735916137976701537177569, −7.47473300714423369691419443631, −6.37290014190301819084305530684, −5.58645560940469331107310999122, −4.62843129750803840444651588062, −3.84904976798826087233266770698, −2.71500331426989487866500015176, −0.02881667057257568834403602662, 2.29665182032762161578606820056, 3.82773730611640363584397226580, 4.88556686510882667471953194632, 5.70817148809054235337887960032, 6.53310027123182151958558469013, 7.11343741359733900855986493162, 8.593712248152724780368740715063, 9.648990621381658494749706225991, 10.46687876882425322691376723214, 11.65254318328670952935990680177

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