Properties

Label 2-525-21.17-c1-0-34
Degree $2$
Conductor $525$
Sign $-0.691 + 0.721i$
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  + (−1.78 + 1.03i)2-s + (0.627 − 1.61i)3-s + (1.12 − 1.95i)4-s + (0.543 + 3.53i)6-s + (0.00953 + 2.64i)7-s + 0.527i·8-s + (−2.21 − 2.02i)9-s + (−4.06 − 2.34i)11-s + (−2.44 − 3.04i)12-s − 0.638i·13-s + (−2.74 − 4.71i)14-s + (1.71 + 2.96i)16-s + (2.07 − 3.59i)17-s + (6.04 + 1.33i)18-s + (−0.776 + 0.448i)19-s + ⋯
L(s)  = 1  + (−1.26 + 0.729i)2-s + (0.362 − 0.932i)3-s + (0.563 − 0.976i)4-s + (0.221 + 1.44i)6-s + (0.00360 + 0.999i)7-s + 0.186i·8-s + (−0.737 − 0.675i)9-s + (−1.22 − 0.707i)11-s + (−0.705 − 0.879i)12-s − 0.177i·13-s + (−0.733 − 1.26i)14-s + (0.427 + 0.741i)16-s + (0.503 − 0.871i)17-s + (1.42 + 0.315i)18-s + (−0.178 + 0.102i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.691 + 0.721i$
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: \((2,\ 525,\ (\ :1/2),\ -0.691 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.103235 - 0.241919i\)
\(L(\frac12)\) \(\approx\) \(0.103235 - 0.241919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.627 + 1.61i)T \)
5 \( 1 \)
7 \( 1 + (-0.00953 - 2.64i)T \)
good2 \( 1 + (1.78 - 1.03i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (4.06 + 2.34i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.638iT - 13T^{2} \)
17 \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.776 - 0.448i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.89 - 3.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.14iT - 29T^{2} \)
31 \( 1 + (2.02 + 1.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.69 + 9.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.10T + 41T^{2} \)
43 \( 1 + 3.14T + 43T^{2} \)
47 \( 1 + (-3.40 - 5.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.96 + 1.13i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.254 - 0.440i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.48 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.41 + 4.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.22iT - 71T^{2} \)
73 \( 1 + (12.5 + 7.22i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.54 + 7.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.76T + 83T^{2} \)
89 \( 1 + (6.90 + 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−10.20370536117373487838457220876, −9.298589977428093008548140705578, −8.571101007441276391570917124691, −7.88278727748237048956218263092, −7.31077843728916635124952813444, −6.07348052205955950464084629214, −5.51760028758756860011316677244, −3.26455627585947485793209503392, −1.99738241235876270162158759337, −0.20988348753053853817253960257, 1.86641077128759345367090850158, 3.12321400451898660851628323824, 4.31129779718580540498510146153, 5.39655365288948191810993169691, 7.08444745785454169433691905131, 8.131809371297749188524393744242, 8.517973179198081254884837734156, 9.852695461572277812779755680406, 10.19872102228009055215904360561, 10.63622716142616278952139185389

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