Properties

Label 2-525-105.23-c1-0-23
Degree $2$
Conductor $525$
Sign $0.424 + 0.905i$
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  + (−2.15 + 0.578i)2-s + (−1.42 + 0.988i)3-s + (2.59 − 1.5i)4-s + (2.5 − 2.95i)6-s + (2.55 + 0.684i)7-s + (−1.58 + 1.58i)8-s + (1.04 − 2.81i)9-s + (−2.21 + 4.70i)12-s + (−3.74 − 3.74i)13-s − 5.91·14-s + (−0.500 + 0.866i)16-s + (−1.15 + 4.31i)17-s + (−0.631 + 6.67i)18-s + (−1.73 − i)19-s + (−4.31 + 1.55i)21-s + ⋯
L(s)  = 1  + (−1.52 + 0.409i)2-s + (−0.821 + 0.570i)3-s + (1.29 − 0.750i)4-s + (1.02 − 1.20i)6-s + (0.965 + 0.258i)7-s + (−0.559 + 0.559i)8-s + (0.348 − 0.937i)9-s + (−0.638 + 1.35i)12-s + (−1.03 − 1.03i)13-s − 1.58·14-s + (−0.125 + 0.216i)16-s + (−0.280 + 1.04i)17-s + (−0.148 + 1.57i)18-s + (−0.397 − 0.229i)19-s + (−0.940 + 0.338i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.424 + 0.905i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.424 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.240034 - 0.152490i\)
\(L(\frac12)\) \(\approx\) \(0.240034 - 0.152490i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.42 - 0.988i)T \)
5 \( 1 \)
7 \( 1 + (-2.55 - 0.684i)T \)
good2 \( 1 + (2.15 - 0.578i)T + (1.73 - i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.74 + 3.74i)T + 13iT^{2} \)
17 \( 1 + (1.15 - 4.31i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.73 + i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 + 6.47i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.91T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.73 + 10.2i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.91iT - 41T^{2} \)
43 \( 1 + (1.87 + 1.87i)T + 43iT^{2} \)
47 \( 1 + (-8.63 + 2.31i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.31 - 1.15i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.91 + 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.55 + 0.684i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (1.36 - 5.11i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.73 + i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.58 - 1.58i)T - 83iT^{2} \)
89 \( 1 + (-2.95 + 5.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.48 - 7.48i)T - 97iT^{2} \)
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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

−10.72174719906040749866041349737, −9.816420972778458681044930265917, −8.941670022067036152208303260291, −8.132530230680447142883108944622, −7.30543065169162663024776116354, −6.22734171153179807140764137279, −5.29574546013992422517292893765, −4.15567521230263079279991420342, −2.05075242819577907843712651699, −0.31940633012054008662272478736, 1.36490059584203482432231622161, 2.29437007544275936298546375319, 4.48878359903851607475367940742, 5.50919478010463426935222394106, 7.18506907784649189981954610804, 7.26293912233020515379532856788, 8.382549182859799367971000067785, 9.334529134289725993466901275545, 10.17021726311980723727216257812, 10.98760195441068092316744609959

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