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Properties

Label 2-525-35.19-c2-0-9
Degree $2$
Conductor $525$
Sign $0.669 - 0.742i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.59 − 1.5i)2-s + (0.866 + 1.5i)3-s + (2.5 + 4.33i)4-s − 5.19i·6-s + (2.59 + 6.5i)7-s − 3.00i·8-s + (−1.5 + 2.59i)9-s + (−7.5 − 12.9i)11-s + (−4.33 + 7.5i)12-s + 13.8·13-s + (3 − 20.7i)14-s + (5.49 − 9.52i)16-s + (5.19 + 9i)17-s + (7.79 − 4.5i)18-s + (9 + 5.19i)19-s + ⋯
L(s)  = 1  + (−1.29 − 0.750i)2-s + (0.288 + 0.5i)3-s + (0.625 + 1.08i)4-s − 0.866i·6-s + (0.371 + 0.928i)7-s − 0.375i·8-s + (−0.166 + 0.288i)9-s + (−0.681 − 1.18i)11-s + (−0.360 + 0.625i)12-s + 1.06·13-s + (0.214 − 1.48i)14-s + (0.343 − 0.595i)16-s + (0.305 + 0.529i)17-s + (0.433 − 0.250i)18-s + (0.473 + 0.273i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.669 - 0.742i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.669 - 0.742i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8926250242\)
\(L(\frac12)\) \(\approx\) \(0.8926250242\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.59 - 6.5i)T \)
good2 \( 1 + (2.59 + 1.5i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (7.5 + 12.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 13.8T + 169T^{2} \)
17 \( 1 + (-5.19 - 9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-9 - 5.19i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 - 9T + 841T^{2} \)
31 \( 1 + (10.5 - 6.06i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (8.66 + 5i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 10.3iT - 1.68e3T^{2} \)
43 \( 1 - 74iT - 1.84e3T^{2} \)
47 \( 1 + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (28.5 - 16.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (13.5 - 7.79i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-78 - 45.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (65.8 - 38i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 84T + 5.04e3T^{2} \)
73 \( 1 + (-31.1 - 54i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (21.5 - 37.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 119.T + 6.88e3T^{2} \)
89 \( 1 + (-63 - 36.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 185.T + 9.40e3T^{2} \)
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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

−10.78555751760146976706904982342, −9.840761675190808099897926360069, −8.990082834930639885939909755360, −8.365372175464559027891717474283, −7.86672989255640156528159228439, −6.08498245523864647629421984554, −5.20837704306401283431260708109, −3.51246605212436729660636470229, −2.59898630791270699676001521541, −1.22955234683668514221609862236, 0.60290859312789126189016881972, 1.83908462682408113037467868626, 3.65037808356275179429924202598, 5.08802319732971381732155064427, 6.44688382356832225839408824773, 7.26362627717817862290623213881, 7.73617841341422350629039274327, 8.583003607838361557641336309354, 9.503689659271710242712788343170, 10.28311162990021250754109297648

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