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Properties

Label 2-525-105.89-c1-0-24
Degree $2$
Conductor $525$
Sign $0.389 - 0.920i$
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  + (−0.866 + 1.5i)2-s + 1.73·3-s + (−0.5 − 0.866i)4-s + (−1.49 + 2.59i)6-s + (0.866 − 2.5i)7-s − 1.73·8-s + 2.99·9-s + (3 − 1.73i)11-s + (−0.866 − 1.50i)12-s + 3.46·13-s + (3 + 3.46i)14-s + (2.49 − 4.33i)16-s + (−5.19 + 3i)17-s + (−2.59 + 4.49i)18-s + (6 + 3.46i)19-s + ⋯
L(s)  = 1  + (−0.612 + 1.06i)2-s + 1.00·3-s + (−0.250 − 0.433i)4-s + (−0.612 + 1.06i)6-s + (0.327 − 0.944i)7-s − 0.612·8-s + 0.999·9-s + (0.904 − 0.522i)11-s + (−0.249 − 0.433i)12-s + 0.960·13-s + (0.801 + 0.925i)14-s + (0.624 − 1.08i)16-s + (−1.26 + 0.727i)17-s + (−0.612 + 1.06i)18-s + (1.37 + 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.389 - 0.920i$
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: \((2,\ 525,\ (\ :1/2),\ 0.389 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35431 + 0.897524i\)
\(L(\frac12)\) \(\approx\) \(1.35431 + 0.897524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
5 \( 1 \)
7 \( 1 + (-0.866 + 2.5i)T \)
good2 \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + (5.19 - 3i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.866 - 1.5i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.73iT - 29T^{2} \)
31 \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 + 2i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.2 - 6.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (1.73 + 3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−10.81243243494814216050591724627, −9.763942764669725085633039885072, −8.909934505269567438574888990740, −8.348807145775785352790919518454, −7.50428485946865767730072012788, −6.80096621583653811954925738727, −5.83025680988162103032916167468, −4.14155564474991405239456537291, −3.35747657652967672257394437758, −1.41238821761972006969970198869, 1.43417768136278236277081248175, 2.44170852374756167251648637033, 3.40872774406492731852483830527, 4.70381502499563334361396306879, 6.20141644342548783305907344934, 7.29566859442582246722274483474, 8.560684904689720400025711065172, 9.124792691876372831671802574375, 9.510405404041829013665197304273, 10.69906400265773073963710743534

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