Properties

Label 2-525-35.33-c1-0-12
Degree $2$
Conductor $525$
Sign $0.952 + 0.304i$
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  + (0.159 + 0.595i)2-s + (−0.965 − 0.258i)3-s + (1.40 − 0.810i)4-s − 0.616i·6-s + (−1.43 − 2.22i)7-s + (1.57 + 1.57i)8-s + (0.866 + 0.499i)9-s + (2.27 + 3.94i)11-s + (−1.56 + 0.419i)12-s + (−0.0478 + 0.0478i)13-s + (1.09 − 1.20i)14-s + (0.932 − 1.61i)16-s + (1.07 − 4.03i)17-s + (−0.159 + 0.595i)18-s + (3.38 − 5.86i)19-s + ⋯
L(s)  = 1  + (0.112 + 0.420i)2-s + (−0.557 − 0.149i)3-s + (0.701 − 0.405i)4-s − 0.251i·6-s + (−0.542 − 0.840i)7-s + (0.557 + 0.557i)8-s + (0.288 + 0.166i)9-s + (0.686 + 1.18i)11-s + (−0.451 + 0.121i)12-s + (−0.0132 + 0.0132i)13-s + (0.292 − 0.323i)14-s + (0.233 − 0.403i)16-s + (0.261 − 0.977i)17-s + (−0.0376 + 0.140i)18-s + (0.776 − 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.952 + 0.304i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.952 + 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53298 - 0.238914i\)
\(L(\frac12)\) \(\approx\) \(1.53298 - 0.238914i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (1.43 + 2.22i)T \)
good2 \( 1 + (-0.159 - 0.595i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-2.27 - 3.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0478 - 0.0478i)T - 13iT^{2} \)
17 \( 1 + (-1.07 + 4.03i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.38 + 5.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.94 + 1.05i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 6.68iT - 29T^{2} \)
31 \( 1 + (2.56 - 1.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.91 - 7.13i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.956iT - 41T^{2} \)
43 \( 1 + (3.47 + 3.47i)T + 43iT^{2} \)
47 \( 1 + (-8.79 + 2.35i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.24 - 12.1i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.09 + 1.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.569 - 0.329i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.1 + 2.72i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.58T + 71T^{2} \)
73 \( 1 + (-7.42 - 1.98i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.84 + 2.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.19 - 7.19i)T - 83iT^{2} \)
89 \( 1 + (7.86 - 13.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.60 - 4.60i)T + 97iT^{2} \)
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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

−10.88327057069706921987945448107, −9.960527661963287223532017733838, −9.309435361539466422033413144244, −7.58758113690216186129247931188, −7.04582866312702891429595400735, −6.48454364268232514331371828894, −5.24662231693657992203165788823, −4.39955025459317799010395841544, −2.73062785298662652215306892040, −1.09483994196229482470533223486, 1.51093356787960936312142454174, 3.13057350159591395318859004201, 3.81793382418834984053803984102, 5.56274556493850062328307837717, 6.13739617526759811458006058294, 7.14666350784349757379196398985, 8.284808714841688105151207650540, 9.227164229118686855128412275552, 10.25405646568035594823601744225, 11.07502832060278243913321324457

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