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Properties

Label 2-525-7.4-c1-0-8
Degree $2$
Conductor $525$
Sign $-0.118 - 0.992i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.776 + 1.34i)2-s + (−0.5 + 0.866i)3-s + (−0.204 + 0.355i)4-s − 1.55·6-s + (2.60 − 0.478i)7-s + 2.46·8-s + (−0.499 − 0.866i)9-s + (−2.21 + 3.83i)11-s + (−0.204 − 0.355i)12-s + 1.73·13-s + (2.66 + 3.12i)14-s + (2.32 + 4.02i)16-s + (1.36 − 2.36i)17-s + (0.776 − 1.34i)18-s + (0.152 + 0.264i)19-s + ⋯
L(s)  = 1  + (0.548 + 0.950i)2-s + (−0.288 + 0.499i)3-s + (−0.102 + 0.177i)4-s − 0.633·6-s + (0.983 − 0.180i)7-s + 0.872·8-s + (−0.166 − 0.288i)9-s + (−0.667 + 1.15i)11-s + (−0.0591 − 0.102i)12-s + 0.480·13-s + (0.711 + 0.835i)14-s + (0.581 + 1.00i)16-s + (0.331 − 0.573i)17-s + (0.182 − 0.316i)18-s + (0.0350 + 0.0606i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.118 - 0.992i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.118 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36435 + 1.53634i\)
\(L(\frac12)\) \(\approx\) \(1.36435 + 1.53634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2.60 + 0.478i)T \)
good2 \( 1 + (-0.776 - 1.34i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (2.21 - 3.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.73T + 13T^{2} \)
17 \( 1 + (-1.36 + 2.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.152 - 0.264i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.51 - 6.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.79T + 29T^{2} \)
31 \( 1 + (-2.64 + 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.83 - 3.18i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.71T + 41T^{2} \)
43 \( 1 + 9.71T + 43T^{2} \)
47 \( 1 + (0.908 + 1.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.857 + 1.48i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.571 + 0.989i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.77 + 8.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.19 + 7.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + (-6.41 + 11.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.35 + 5.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.09T + 83T^{2} \)
89 \( 1 + (2.03 + 3.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.87T + 97T^{2} \)
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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

−11.11505212730342867966626455952, −10.24180060007434475997562488874, −9.410270438862746747358964506691, −8.005520370739228261612583443104, −7.46140059545576093463923891912, −6.45021798959771280376205948298, −5.13652358228410449942079433923, −5.02475928111298576121091323645, −3.72632806228961611230493363588, −1.75562157789578956132155190117, 1.26553505576720249603792869300, 2.50591571573454311537310015952, 3.62945550904974858246322930363, 4.88565208534300547621010569460, 5.70418536588492488043214229447, 6.98676775105873400214070027257, 8.095449328599591013849474162117, 8.591661442691858224679214278875, 10.24951236361186574666718742725, 11.02208290278601512602802715914

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