Properties

Label 2-525-5.4-c3-0-55
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  − 3.53i·2-s − 3i·3-s − 4.46·4-s − 10.5·6-s − 7i·7-s − 12.4i·8-s − 9·9-s − 2.93·11-s + 13.4i·12-s + 19.0i·13-s − 24.7·14-s − 79.7·16-s + 122. i·17-s + 31.7i·18-s − 107.·19-s + ⋯
L(s)  = 1  − 1.24i·2-s − 0.577i·3-s − 0.558·4-s − 0.720·6-s − 0.377i·7-s − 0.551i·8-s − 0.333·9-s − 0.0805·11-s + 0.322i·12-s + 0.406i·13-s − 0.471·14-s − 1.24·16-s + 1.74i·17-s + 0.416i·18-s − 1.29·19-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

\(L(2)\) \(\approx\) \(0.2748148176\)
\(L(\frac12)\) \(\approx\) \(0.2748148176\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.53iT - 8T^{2} \)
11 \( 1 + 2.93T + 1.33e3T^{2} \)
13 \( 1 - 19.0iT - 2.19e3T^{2} \)
17 \( 1 - 122. iT - 4.91e3T^{2} \)
19 \( 1 + 107.T + 6.85e3T^{2} \)
23 \( 1 + 210. iT - 1.21e4T^{2} \)
29 \( 1 + 95.4T + 2.43e4T^{2} \)
31 \( 1 + 94.3T + 2.97e4T^{2} \)
37 \( 1 - 97.1iT - 5.06e4T^{2} \)
41 \( 1 + 491.T + 6.89e4T^{2} \)
43 \( 1 - 43.0iT - 7.95e4T^{2} \)
47 \( 1 - 473. iT - 1.03e5T^{2} \)
53 \( 1 - 183. iT - 1.48e5T^{2} \)
59 \( 1 - 760.T + 2.05e5T^{2} \)
61 \( 1 + 198.T + 2.26e5T^{2} \)
67 \( 1 + 309. iT - 3.00e5T^{2} \)
71 \( 1 - 665.T + 3.57e5T^{2} \)
73 \( 1 + 621. iT - 3.89e5T^{2} \)
79 \( 1 - 24.7T + 4.93e5T^{2} \)
83 \( 1 - 406. iT - 5.71e5T^{2} \)
89 \( 1 + 261.T + 7.04e5T^{2} \)
97 \( 1 + 1.00e3iT - 9.12e5T^{2} \)
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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

−10.13267327480511023688443053954, −8.884484148176700563313969444454, −8.141235261503905465757917128786, −6.81884674080411514778326676708, −6.21815021159355175710758615853, −4.52290368930383097628961529034, −3.64905251990573294445280500922, −2.34224726254288258447271356778, −1.50147145185159353282173626839, −0.07812670829720184653250556573, 2.25852779927536283359697924654, 3.65464113277708188772499867254, 5.12339337988887692799664641898, 5.47646647078565872685525504207, 6.70563780339687115985473973071, 7.46468960638725468710410316764, 8.444791377098424965467956167625, 9.177946855711492767613530452866, 10.08166209625501934692489172243, 11.25973190252054339547365832034

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