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Properties

Label 2-525-5.4-c3-0-6
Degree $2$
Conductor $525$
Sign $-0.447 + 0.894i$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.70i·2-s − 3i·3-s − 14.1·4-s + 14.1·6-s + 7i·7-s − 28.7i·8-s − 9·9-s + 24.5·11-s + 42.3i·12-s + 35.0i·13-s − 32.9·14-s + 22.1·16-s − 18.4i·17-s − 42.3i·18-s + 67.4·19-s + ⋯
L(s)  = 1  + 1.66i·2-s − 0.577i·3-s − 1.76·4-s + 0.959·6-s + 0.377i·7-s − 1.26i·8-s − 0.333·9-s + 0.674·11-s + 1.01i·12-s + 0.747i·13-s − 0.628·14-s + 0.345·16-s − 0.262i·17-s − 0.554i·18-s + 0.813·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7114551850\)
\(L(\frac12)\) \(\approx\) \(0.7114551850\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 \)
7 \( 1 - 7iT \)
good2 \( 1 - 4.70iT - 8T^{2} \)
11 \( 1 - 24.5T + 1.33e3T^{2} \)
13 \( 1 - 35.0iT - 2.19e3T^{2} \)
17 \( 1 + 18.4iT - 4.91e3T^{2} \)
19 \( 1 - 67.4T + 6.85e3T^{2} \)
23 \( 1 - 145. iT - 1.21e4T^{2} \)
29 \( 1 + 214.T + 2.43e4T^{2} \)
31 \( 1 + 88.6T + 2.97e4T^{2} \)
37 \( 1 - 162. iT - 5.06e4T^{2} \)
41 \( 1 + 337.T + 6.89e4T^{2} \)
43 \( 1 + 122. iT - 7.95e4T^{2} \)
47 \( 1 - 354. iT - 1.03e5T^{2} \)
53 \( 1 + 676. iT - 1.48e5T^{2} \)
59 \( 1 + 501.T + 2.05e5T^{2} \)
61 \( 1 + 708.T + 2.26e5T^{2} \)
67 \( 1 + 907. iT - 3.00e5T^{2} \)
71 \( 1 - 430.T + 3.57e5T^{2} \)
73 \( 1 + 41.3iT - 3.89e5T^{2} \)
79 \( 1 + 890.T + 4.93e5T^{2} \)
83 \( 1 - 1.05e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.47e3T + 7.04e5T^{2} \)
97 \( 1 - 555. iT - 9.12e5T^{2} \)
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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

−11.27739763172448817903954395878, −9.529287500260280904841921732733, −9.124070215834131592812891800468, −8.060201619353959982714263624243, −7.32096340095898246774710912719, −6.61047134681383072474328247354, −5.74711471816309571751595919773, −4.91664470153598504592470692001, −3.54077301345161542454763528334, −1.66175388036912982330645776556, 0.22118455616605298102314122836, 1.53153012872217405322416020043, 2.90805334431274191067107390342, 3.76250317392425434015619871047, 4.60491514471741011187902883542, 5.78314586977567984563206079364, 7.25097390259322107168228324747, 8.557009241930014478857564783659, 9.309617137185065630889621205172, 10.11742334010186195950749351207

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