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Properties

Label 2-525-21.17-c1-0-44
Degree $2$
Conductor $525$
Sign $-0.417 + 0.908i$
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  + (1.94 − 1.12i)2-s + (0.983 − 1.42i)3-s + (1.53 − 2.65i)4-s + (0.313 − 3.88i)6-s + (−1.42 − 2.22i)7-s − 2.39i·8-s + (−1.06 − 2.80i)9-s + (1.64 + 0.952i)11-s + (−2.27 − 4.79i)12-s + 5.07i·13-s + (−5.29 − 2.73i)14-s + (0.369 + 0.639i)16-s + (−2.22 + 3.85i)17-s + (−5.22 − 4.26i)18-s + (3.85 − 2.22i)19-s + ⋯
L(s)  = 1  + (1.37 − 0.795i)2-s + (0.568 − 0.822i)3-s + (0.766 − 1.32i)4-s + (0.128 − 1.58i)6-s + (−0.540 − 0.841i)7-s − 0.846i·8-s + (−0.354 − 0.935i)9-s + (0.497 + 0.287i)11-s + (−0.656 − 1.38i)12-s + 1.40i·13-s + (−1.41 − 0.730i)14-s + (0.0923 + 0.159i)16-s + (−0.540 + 0.936i)17-s + (−1.23 − 1.00i)18-s + (0.884 − 0.510i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.417 + 0.908i$
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: \((2,\ 525,\ (\ :1/2),\ -0.417 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80582 - 2.81627i\)
\(L(\frac12)\) \(\approx\) \(1.80582 - 2.81627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.983 + 1.42i)T \)
5 \( 1 \)
7 \( 1 + (1.42 + 2.22i)T \)
good2 \( 1 + (-1.94 + 1.12i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-1.64 - 0.952i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.07iT - 13T^{2} \)
17 \( 1 + (2.22 - 3.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.85 + 2.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.46 - 1.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.82iT - 29T^{2} \)
31 \( 1 + (-4.81 - 2.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.32 + 4.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.250T + 41T^{2} \)
43 \( 1 - 9.23T + 43T^{2} \)
47 \( 1 + (1.53 + 2.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.09 + 1.21i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.95 - 12.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.51 - 0.874i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.14 - 7.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.68iT - 71T^{2} \)
73 \( 1 + (5.47 + 3.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.59 - 2.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.98T + 83T^{2} \)
89 \( 1 + (-5.43 - 9.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.94iT - 97T^{2} \)
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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

−10.93048437951193540466504449509, −9.826635973840775856061057781435, −8.939759145116969744327704887239, −7.62849809607702197443868143156, −6.67068143387580483204964028885, −6.01285540904762366511674515043, −4.35892895005499837207761295909, −3.82754344759723892442107940328, −2.59707844515758704758832779257, −1.47831515164954384335036962303, 2.85756608975510196198524094972, 3.39421334878478715870357996479, 4.65907093737894178058103014483, 5.44510038551878594840598005688, 6.20121400004083472216931848818, 7.40735505409601580297606102912, 8.342914282359519728370834652732, 9.322026440508592829725344879297, 10.14621613842555608903775407728, 11.34277979009983225041314371020

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