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Properties

Label 2-525-105.23-c1-0-15
Degree $2$
Conductor $525$
Sign $0.580 - 0.814i$
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  + (−0.907 + 0.243i)2-s + (−1.12 + 1.31i)3-s + (−0.967 + 0.558i)4-s + (0.700 − 1.46i)6-s + (2.64 + 0.0144i)7-s + (2.07 − 2.07i)8-s + (−0.470 − 2.96i)9-s + (0.630 − 0.363i)11-s + (0.352 − 1.90i)12-s + (1.44 + 1.44i)13-s + (−2.40 + 0.630i)14-s + (−0.257 + 0.446i)16-s + (1.90 − 7.09i)17-s + (1.14 + 2.57i)18-s + (0.664 + 0.383i)19-s + ⋯
L(s)  = 1  + (−0.641 + 0.171i)2-s + (−0.649 + 0.760i)3-s + (−0.483 + 0.279i)4-s + (0.285 − 0.599i)6-s + (0.999 + 0.00544i)7-s + (0.732 − 0.732i)8-s + (−0.156 − 0.987i)9-s + (0.189 − 0.109i)11-s + (0.101 − 0.549i)12-s + (0.400 + 0.400i)13-s + (−0.642 + 0.168i)14-s + (−0.0644 + 0.111i)16-s + (0.460 − 1.71i)17-s + (0.270 + 0.606i)18-s + (0.152 + 0.0879i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.580 - 0.814i$
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: \((2,\ 525,\ (\ :1/2),\ 0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.731983 + 0.377390i\)
\(L(\frac12)\) \(\approx\) \(0.731983 + 0.377390i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.12 - 1.31i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 - 0.0144i)T \)
good2 \( 1 + (0.907 - 0.243i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-0.630 + 0.363i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.44 - 1.44i)T + 13iT^{2} \)
17 \( 1 + (-1.90 + 7.09i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.664 - 0.383i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.840 - 3.13i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 4.07T + 29T^{2} \)
31 \( 1 + (0.209 + 0.363i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.63 - 6.08i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.44iT - 41T^{2} \)
43 \( 1 + (-5.15 - 5.15i)T + 43iT^{2} \)
47 \( 1 + (6.79 - 1.82i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.26 + 1.41i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.807 - 1.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.78 + 8.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.90 - 1.84i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.06iT - 71T^{2} \)
73 \( 1 + (-4.08 + 15.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.80 - 3.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.83 - 1.83i)T - 83iT^{2} \)
89 \( 1 + (6.94 - 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.62 - 5.62i)T - 97iT^{2} \)
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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

−11.03877029447670141105513124670, −9.824449431916446182934073386502, −9.407884519390653862042213365360, −8.426408894878498032468333448594, −7.57964266224393790214718993868, −6.48153035150786786351369407794, −5.10310428486162121406226981232, −4.56868665657349610573441056643, −3.33140180249513143478981239270, −1.02262994091194370085486783220, 0.977830523046016512502142078742, 2.02383720366068962950283168192, 4.16908492171630734585998615082, 5.25233393452087335850146249836, 6.01985551348350417068748003166, 7.30210376878518475273753183241, 8.249471376099446741234907479243, 8.636102351779579425936670821119, 10.08955982587483344313904484424, 10.72692303317481079806151684498

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