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Properties

Label 2-525-5.4-c3-0-20
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  − 4.53i·2-s + 3i·3-s − 12.5·4-s + 13.5·6-s + 7i·7-s + 20.5i·8-s − 9·9-s − 19.0·11-s − 37.5i·12-s − 2.93i·13-s + 31.7·14-s − 7.21·16-s + 6.49i·17-s + 40.7i·18-s + 5.43·19-s + ⋯
L(s)  = 1  − 1.60i·2-s + 0.577i·3-s − 1.56·4-s + 0.924·6-s + 0.377i·7-s + 0.907i·8-s − 0.333·9-s − 0.522·11-s − 0.904i·12-s − 0.0626i·13-s + 0.605·14-s − 0.112·16-s + 0.0927i·17-s + 0.533i·18-s + 0.0656·19-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

\(L(2)\) \(\approx\) \(1.595828317\)
\(L(\frac12)\) \(\approx\) \(1.595828317\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.53iT - 8T^{2} \)
11 \( 1 + 19.0T + 1.33e3T^{2} \)
13 \( 1 + 2.93iT - 2.19e3T^{2} \)
17 \( 1 - 6.49iT - 4.91e3T^{2} \)
19 \( 1 - 5.43T + 6.85e3T^{2} \)
23 \( 1 - 49.3iT - 1.21e4T^{2} \)
29 \( 1 - 291.T + 2.43e4T^{2} \)
31 \( 1 - 244.T + 2.97e4T^{2} \)
37 \( 1 - 193. iT - 5.06e4T^{2} \)
41 \( 1 - 315.T + 6.89e4T^{2} \)
43 \( 1 + 300. iT - 7.95e4T^{2} \)
47 \( 1 + 86.5iT - 1.03e5T^{2} \)
53 \( 1 - 509. iT - 1.48e5T^{2} \)
59 \( 1 - 83.3T + 2.05e5T^{2} \)
61 \( 1 + 5.25T + 2.26e5T^{2} \)
67 \( 1 + 205. iT - 3.00e5T^{2} \)
71 \( 1 - 1.00e3T + 3.57e5T^{2} \)
73 \( 1 + 1.00e3iT - 3.89e5T^{2} \)
79 \( 1 - 863.T + 4.93e5T^{2} \)
83 \( 1 - 1.33e3iT - 5.71e5T^{2} \)
89 \( 1 + 326.T + 7.04e5T^{2} \)
97 \( 1 + 1.52e3iT - 9.12e5T^{2} \)
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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

−10.37886639460685881227761049474, −9.757080419167479103519571713321, −8.887331437523068612966163298906, −8.016587974664946570600668043427, −6.45658335329951133032491410051, −5.15095990825133038978404529576, −4.31358716731988401315353683963, −3.16997452289445418459853478361, −2.37717022641922166950124653938, −0.868343270078126560103328904762, 0.69665451496178968429292956999, 2.62292098925399939831972819075, 4.33430164799673217844166853276, 5.24789212667538642918090425330, 6.31714700248074349796356714708, 6.85758866460490217686195210065, 7.891179616727495944599954178332, 8.301017886816017981260461214780, 9.401933122833128782539599914653, 10.45329260520514514942915328639

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