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Properties

Label 2-525-21.17-c1-0-5
Degree $2$
Conductor $525$
Sign $-0.909 - 0.415i$
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  + (0.766 − 0.442i)2-s + (−0.189 + 1.72i)3-s + (−0.608 + 1.05i)4-s + (0.616 + 1.40i)6-s + (−2.63 + 0.206i)7-s + 2.84i·8-s + (−2.92 − 0.652i)9-s + (1.25 + 0.723i)11-s + (−1.69 − 1.24i)12-s − 4.04i·13-s + (−1.93 + 1.32i)14-s + (0.0422 + 0.0730i)16-s + (−2.87 + 4.98i)17-s + (−2.53 + 0.795i)18-s + (−0.356 + 0.206i)19-s + ⋯
L(s)  = 1  + (0.541 − 0.312i)2-s + (−0.109 + 0.993i)3-s + (−0.304 + 0.527i)4-s + (0.251 + 0.572i)6-s + (−0.996 + 0.0778i)7-s + 1.00i·8-s + (−0.976 − 0.217i)9-s + (0.377 + 0.218i)11-s + (−0.490 − 0.360i)12-s − 1.12i·13-s + (−0.515 + 0.354i)14-s + (0.0105 + 0.0182i)16-s + (−0.698 + 1.20i)17-s + (−0.596 + 0.187i)18-s + (−0.0818 + 0.0472i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.909 - 0.415i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.909 - 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.196040 + 0.900947i\)
\(L(\frac12)\) \(\approx\) \(0.196040 + 0.900947i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.189 - 1.72i)T \)
5 \( 1 \)
7 \( 1 + (2.63 - 0.206i)T \)
good2 \( 1 + (-0.766 + 0.442i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-1.25 - 0.723i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.04iT - 13T^{2} \)
17 \( 1 + (2.87 - 4.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.356 - 0.206i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.33 - 3.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.82iT - 29T^{2} \)
31 \( 1 + (-2.30 - 1.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.92 - 5.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.46T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + (-5.43 - 9.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.20 - 0.697i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.583 - 1.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.58 + 2.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 - 5.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 + (-2.72 - 1.57i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.42 - 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + (3.90 + 6.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.86iT - 97T^{2} \)
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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

−11.28608835693090244414360003799, −10.31180241241951236265583511016, −9.647612426761695384596415225289, −8.657603401148441381559779057080, −7.907594204712084283007750520516, −6.32326982691035690245505592383, −5.53171481479418501171434330284, −4.29731163742173106851267852624, −3.65317315532502138978653860157, −2.66035669818498545417223305339, 0.44355132078063413667157093853, 2.21558680213892171105529766319, 3.75752615484339190019338469707, 4.90590031742065750448347027791, 6.17692142632567042673904046796, 6.52537690804173368822175613002, 7.39484972229336227829374398424, 8.831839745132067832474920710487, 9.429593011754728975381702798395, 10.48970734826314234262148057830

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