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Properties

Label 2-525-25.14-c1-0-9
Degree $2$
Conductor $525$
Sign $-0.408 + 0.912i$
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  + (−1.36 − 1.87i)2-s + (−0.951 − 0.309i)3-s + (−1.04 + 3.21i)4-s + (−1.94 + 1.09i)5-s + (0.717 + 2.20i)6-s + i·7-s + (3.05 − 0.993i)8-s + (0.809 + 0.587i)9-s + (4.71 + 2.16i)10-s + (−4.18 + 3.04i)11-s + (1.98 − 2.73i)12-s + (2.62 − 3.61i)13-s + (1.87 − 1.36i)14-s + (2.19 − 0.440i)15-s + (−0.556 − 0.404i)16-s + (2.52 − 0.821i)17-s + ⋯
L(s)  = 1  + (−0.964 − 1.32i)2-s + (−0.549 − 0.178i)3-s + (−0.523 + 1.60i)4-s + (−0.871 + 0.490i)5-s + (0.292 + 0.900i)6-s + 0.377i·7-s + (1.08 − 0.351i)8-s + (0.269 + 0.195i)9-s + (1.49 + 0.684i)10-s + (−1.26 + 0.917i)11-s + (0.574 − 0.790i)12-s + (0.728 − 1.00i)13-s + (0.501 − 0.364i)14-s + (0.566 − 0.113i)15-s + (−0.139 − 0.101i)16-s + (0.613 − 0.199i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.408 + 0.912i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245130 - 0.378166i\)
\(L(\frac12)\) \(\approx\) \(0.245130 - 0.378166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (1.94 - 1.09i)T \)
7 \( 1 - iT \)
good2 \( 1 + (1.36 + 1.87i)T + (-0.618 + 1.90i)T^{2} \)
11 \( 1 + (4.18 - 3.04i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.62 + 3.61i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.52 + 0.821i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.867 + 2.66i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.553 + 0.761i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.57 + 4.85i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.74 - 5.36i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.39 + 6.05i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.27 + 0.928i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.000338iT - 43T^{2} \)
47 \( 1 + (-8.04 - 2.61i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.52 + 1.79i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.94 - 7.22i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.69 - 5.59i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.64 - 0.533i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-4.05 + 12.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.76 + 2.42i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.83 + 5.63i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-8.07 + 2.62i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-14.8 + 10.7i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-8.70 - 2.82i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.53933594667819576525155341876, −10.16909827184062198944475678439, −8.927264961561111119073398217078, −7.960968542250965244747651487253, −7.43492787634320735956131631070, −5.94846635584634484134393517441, −4.63286819014118924084127814253, −3.26656819645628031208185196909, −2.35597666359479465062480679398, −0.56303470137742133775512426355, 0.900455193625010494317581568973, 3.63191443058749882262338763331, 4.90080817059457192634569642360, 5.82074597732981309581208093266, 6.67797236161502742774453267108, 7.77353421310401908980543220909, 8.205458806623131289495122365494, 9.090520154833372373233981110936, 10.11517333850348370742644165016, 10.92264831003374362531975506135

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