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Properties

Label 2-525-5.3-c2-0-12
Degree $2$
Conductor $525$
Sign $0.850 - 0.525i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.41 + 2.41i)2-s + (−1.22 − 1.22i)3-s − 7.68i·4-s + 5.92·6-s + (1.87 − 1.87i)7-s + (8.90 + 8.90i)8-s + 2.99i·9-s − 20.9·11-s + (−9.40 + 9.40i)12-s + (9.34 + 9.34i)13-s + 9.04i·14-s − 12.2·16-s + (−7.08 + 7.08i)17-s + (−7.25 − 7.25i)18-s + 14.9i·19-s + ⋯
L(s)  = 1  + (−1.20 + 1.20i)2-s + (−0.408 − 0.408i)3-s − 1.92i·4-s + 0.986·6-s + (0.267 − 0.267i)7-s + (1.11 + 1.11i)8-s + 0.333i·9-s − 1.90·11-s + (−0.784 + 0.784i)12-s + (0.718 + 0.718i)13-s + 0.645i·14-s − 0.768·16-s + (−0.416 + 0.416i)17-s + (−0.402 − 0.402i)18-s + 0.786i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5815661401\)
\(L(\frac12)\) \(\approx\) \(0.5815661401\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
7 \( 1 + (-1.87 + 1.87i)T \)
good2 \( 1 + (2.41 - 2.41i)T - 4iT^{2} \)
11 \( 1 + 20.9T + 121T^{2} \)
13 \( 1 + (-9.34 - 9.34i)T + 169iT^{2} \)
17 \( 1 + (7.08 - 7.08i)T - 289iT^{2} \)
19 \( 1 - 14.9iT - 361T^{2} \)
23 \( 1 + (12.9 + 12.9i)T + 529iT^{2} \)
29 \( 1 + 39.6iT - 841T^{2} \)
31 \( 1 - 12.8T + 961T^{2} \)
37 \( 1 + (-31.7 + 31.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 69.4T + 1.68e3T^{2} \)
43 \( 1 + (4.46 + 4.46i)T + 1.84e3iT^{2} \)
47 \( 1 + (-4.41 + 4.41i)T - 2.20e3iT^{2} \)
53 \( 1 + (-48.5 - 48.5i)T + 2.80e3iT^{2} \)
59 \( 1 + 29.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.09T + 3.72e3T^{2} \)
67 \( 1 + (-1.39 + 1.39i)T - 4.48e3iT^{2} \)
71 \( 1 + 15.9T + 5.04e3T^{2} \)
73 \( 1 + (32.4 + 32.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 66.1iT - 6.24e3T^{2} \)
83 \( 1 + (-83.6 - 83.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 62.7iT - 7.92e3T^{2} \)
97 \( 1 + (-85.4 + 85.4i)T - 9.40e3iT^{2} \)
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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

−10.52765750453583437920049854983, −9.774703284990081245669362936532, −8.599184561429635529335593091334, −7.935233952130525458921209635220, −7.37928999032059536033986855629, −6.19754661069986421396162669124, −5.71450720452280620616189494137, −4.37856805070432028656048959018, −2.16307135966141551887193232599, −0.58077441147850556631629640996, 0.73679554414560136098856792847, 2.38242066884260246616263539475, 3.25362355620525678111995351943, 4.78028868838572821663467539389, 5.79414522817956976029696136987, 7.42106852604399436931293188735, 8.202392513845794717452226278656, 8.980660746602395811662179618144, 9.922408042614545574387936057668, 10.63039894925255894426684176959

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