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Properties

Label 2-525-105.89-c1-0-38
Degree $2$
Conductor $525$
Sign $-0.946 + 0.321i$
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  + (−0.539 + 0.933i)2-s + (−0.0613 − 1.73i)3-s + (0.418 + 0.725i)4-s + (1.64 + 0.876i)6-s + (−0.929 − 2.47i)7-s − 3.05·8-s + (−2.99 + 0.212i)9-s + (−3.84 + 2.21i)11-s + (1.22 − 0.769i)12-s − 0.955·13-s + (2.81 + 0.467i)14-s + (0.812 − 1.40i)16-s + (−0.439 + 0.253i)17-s + (1.41 − 2.90i)18-s + (−4.41 − 2.54i)19-s + ⋯
L(s)  = 1  + (−0.381 + 0.660i)2-s + (−0.0354 − 0.999i)3-s + (0.209 + 0.362i)4-s + (0.673 + 0.357i)6-s + (−0.351 − 0.936i)7-s − 1.08·8-s + (−0.997 + 0.0707i)9-s + (−1.15 + 0.669i)11-s + (0.354 − 0.222i)12-s − 0.265·13-s + (0.752 + 0.125i)14-s + (0.203 − 0.351i)16-s + (−0.106 + 0.0615i)17-s + (0.333 − 0.685i)18-s + (−1.01 − 0.584i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.946 + 0.321i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.946 + 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0112598 - 0.0681499i\)
\(L(\frac12)\) \(\approx\) \(0.0112598 - 0.0681499i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0613 + 1.73i)T \)
5 \( 1 \)
7 \( 1 + (0.929 + 2.47i)T \)
good2 \( 1 + (0.539 - 0.933i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (3.84 - 2.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.955T + 13T^{2} \)
17 \( 1 + (0.439 - 0.253i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.41 + 2.54i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.14 - 3.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.89iT - 29T^{2} \)
31 \( 1 + (-5.10 + 2.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.51 + 3.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.65T + 41T^{2} \)
43 \( 1 - 0.492iT - 43T^{2} \)
47 \( 1 + (5.76 + 3.32i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.56 + 7.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.81 + 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.399 - 0.230i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.20 - 1.85i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.90iT - 71T^{2} \)
73 \( 1 + (-3.15 - 5.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.38 + 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.7iT - 83T^{2} \)
89 \( 1 + (3.57 - 6.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.91T + 97T^{2} \)
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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

−10.49507040297602534624480673913, −9.374344699925693756081696506527, −8.281407198024252942964539946509, −7.64558308114631261972748746378, −6.98164198159013004250194498134, −6.29327056958955699273636749316, −5.00403389624996172929378275735, −3.38821653200816151624149420670, −2.17816952677805709240739931309, −0.04091054143330528458745910865, 2.36546054587351611505771946556, 3.10352364959795077054072906376, 4.63080296068699546915733235053, 5.75245174459538448870540580872, 6.27589170755123430319443268242, 8.175751883379721101174365532836, 8.788040161263165831977692988663, 9.760843862656359694060481123499, 10.34509417370971908251043097874, 10.98564627650493120056330510603

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