Properties

Label 2-525-5.4-c3-0-37
Degree $2$
Conductor $525$
Sign $0.894 + 0.447i$
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  + 1.37i·2-s − 3i·3-s + 6.10·4-s + 4.12·6-s + 7i·7-s + 19.4i·8-s − 9·9-s + 15.2·11-s − 18.3i·12-s − 76.7i·13-s − 9.63·14-s + 22.1·16-s − 96.7i·17-s − 12.3i·18-s + 14.1·19-s + ⋯
L(s)  = 1  + 0.486i·2-s − 0.577i·3-s + 0.763·4-s + 0.280·6-s + 0.377i·7-s + 0.857i·8-s − 0.333·9-s + 0.418·11-s − 0.440i·12-s − 1.63i·13-s − 0.183·14-s + 0.345·16-s − 1.37i·17-s − 0.162i·18-s + 0.171·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
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: \((2,\ 525,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.461987953\)
\(L(\frac12)\) \(\approx\) \(2.461987953\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.37iT - 8T^{2} \)
11 \( 1 - 15.2T + 1.33e3T^{2} \)
13 \( 1 + 76.7iT - 2.19e3T^{2} \)
17 \( 1 + 96.7iT - 4.91e3T^{2} \)
19 \( 1 - 14.1T + 6.85e3T^{2} \)
23 \( 1 - 75.7iT - 1.21e4T^{2} \)
29 \( 1 + 89.9T + 2.43e4T^{2} \)
31 \( 1 - 289.T + 2.97e4T^{2} \)
37 \( 1 - 14.2iT - 5.06e4T^{2} \)
41 \( 1 - 318.T + 6.89e4T^{2} \)
43 \( 1 + 389. iT - 7.95e4T^{2} \)
47 \( 1 - 228. iT - 1.03e5T^{2} \)
53 \( 1 + 679. iT - 1.48e5T^{2} \)
59 \( 1 - 398.T + 2.05e5T^{2} \)
61 \( 1 + 146.T + 2.26e5T^{2} \)
67 \( 1 - 291. iT - 3.00e5T^{2} \)
71 \( 1 - 333.T + 3.57e5T^{2} \)
73 \( 1 + 891. iT - 3.89e5T^{2} \)
79 \( 1 - 416.T + 4.93e5T^{2} \)
83 \( 1 - 814. iT - 5.71e5T^{2} \)
89 \( 1 - 650.T + 7.04e5T^{2} \)
97 \( 1 + 1.58e3iT - 9.12e5T^{2} \)
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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

−10.46283642104294008107252572739, −9.416486868471503021406790625721, −8.262804451114877317583255377168, −7.61687084098038135321477489561, −6.80437893050605341958058842295, −5.84168714831764970580184769516, −5.13560336206100503910500276209, −3.24226241701827172452644867538, −2.33362046304417805895279085841, −0.816391722085250691354384788376, 1.26540969360033535000070336614, 2.45602196428391404100093843614, 3.79240355175365033152429435516, 4.45140054116908207450207521880, 6.10663827929117703583051515354, 6.68719307144106017966583492903, 7.84224071745513322243202703381, 8.967126719455028923366501743529, 9.819112557675717858328101948638, 10.59558394431659182323138815576

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