Properties

Label 2-525-105.59-c1-0-39
Degree $2$
Conductor $525$
Sign $-0.245 - 0.969i$
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  + (−1.16 − 2.01i)2-s + (0.436 − 1.67i)3-s + (−1.71 + 2.97i)4-s + (−3.89 + 1.07i)6-s + (2.39 − 1.11i)7-s + 3.33·8-s + (−2.61 − 1.46i)9-s + (−2.42 − 1.39i)11-s + (4.23 + 4.17i)12-s − 3.20·13-s + (−5.04 − 3.53i)14-s + (−0.459 − 0.795i)16-s + (−0.763 − 0.440i)17-s + (0.0942 + 6.99i)18-s + (−1.90 + 1.09i)19-s + ⋯
L(s)  = 1  + (−0.824 − 1.42i)2-s + (0.252 − 0.967i)3-s + (−0.858 + 1.48i)4-s + (−1.58 + 0.437i)6-s + (0.906 − 0.422i)7-s + 1.18·8-s + (−0.872 − 0.488i)9-s + (−0.729 − 0.421i)11-s + (1.22 + 1.20i)12-s − 0.888·13-s + (−1.34 − 0.946i)14-s + (−0.114 − 0.198i)16-s + (−0.185 − 0.106i)17-s + (0.0222 + 1.64i)18-s + (−0.436 + 0.251i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.245 - 0.969i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.245 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.353094 + 0.453455i\)
\(L(\frac12)\) \(\approx\) \(0.353094 + 0.453455i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.436 + 1.67i)T \)
5 \( 1 \)
7 \( 1 + (-2.39 + 1.11i)T \)
good2 \( 1 + (1.16 + 2.01i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (2.42 + 1.39i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.20T + 13T^{2} \)
17 \( 1 + (0.763 + 0.440i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.90 - 1.09i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.77 + 6.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.15iT - 29T^{2} \)
31 \( 1 + (7.62 + 4.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.352 - 0.203i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.55T + 41T^{2} \)
43 \( 1 + 0.118iT - 43T^{2} \)
47 \( 1 + (-2.27 + 1.31i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.73 + 6.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.04 + 3.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.7 + 6.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.38 + 0.802i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.25iT - 71T^{2} \)
73 \( 1 + (0.110 - 0.192i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.56 + 2.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.666iT - 83T^{2} \)
89 \( 1 + (0.437 + 0.757i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.37T + 97T^{2} \)
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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

−10.49815778240173617968885394377, −9.385097354801705357755938436038, −8.441313093384884072451770003036, −7.942573564613574970515325459222, −6.95493185187535902924487737085, −5.45232649429870501002035874351, −3.99712993817355009436604736732, −2.64540580733690255380703542153, −1.86810088450147691338128957870, −0.40879800416285270259592670552, 2.37143474763736354307504059389, 4.26256800936823614329280583093, 5.25509745872725638553392354108, 5.82813215131111830641163222831, 7.36942270694535324334634961467, 7.87619128972870886219400575472, 8.760040572001180427634643353979, 9.481387675141112040118011539372, 10.19851337002704092950430978287, 11.14242144943371121388982732721

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