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Properties

Label 2-525-35.19-c2-0-15
Degree $2$
Conductor $525$
Sign $0.657 + 0.753i$
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  + (−0.866 − 0.5i)2-s + (−0.866 − 1.5i)3-s + (−1.5 − 2.59i)4-s + 1.73i·6-s + (−4.33 + 5.5i)7-s + 7i·8-s + (−1.5 + 2.59i)9-s + (−2 − 3.46i)11-s + (−2.59 + 4.5i)12-s + 17.3·13-s + (6.5 − 2.59i)14-s + (−2.5 + 4.33i)16-s + (4.33 + 7.5i)17-s + (2.59 − 1.5i)18-s + (−6 − 3.46i)19-s + ⋯
L(s)  = 1  + (−0.433 − 0.250i)2-s + (−0.288 − 0.5i)3-s + (−0.375 − 0.649i)4-s + 0.288i·6-s + (−0.618 + 0.785i)7-s + 0.875i·8-s + (−0.166 + 0.288i)9-s + (−0.181 − 0.314i)11-s + (−0.216 + 0.375i)12-s + 1.33·13-s + (0.464 − 0.185i)14-s + (−0.156 + 0.270i)16-s + (0.254 + 0.441i)17-s + (0.144 − 0.0833i)18-s + (−0.315 − 0.182i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.657 + 0.753i$
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: \((2,\ 525,\ (\ :1),\ 0.657 + 0.753i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9837839286\)
\(L(\frac12)\) \(\approx\) \(0.9837839286\)
\(L(2)\) 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 \( 1 + (4.33 - 5.5i)T \)
good2 \( 1 + (0.866 + 0.5i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 17.3T + 169T^{2} \)
17 \( 1 + (-4.33 - 7.5i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (6 + 3.46i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-26.8 - 15.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 10T + 841T^{2} \)
31 \( 1 + (-25.5 + 14.7i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (43.3 + 25i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 53.6iT - 1.68e3T^{2} \)
43 \( 1 - 34iT - 1.84e3T^{2} \)
47 \( 1 + (21.6 - 37.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-43.3 + 25i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-48 + 27.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-21 - 12.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-43.3 + 25i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 97T + 5.04e3T^{2} \)
73 \( 1 + (24.2 + 42i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (3.5 - 6.06i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 152.T + 6.88e3T^{2} \)
89 \( 1 + (-136.5 - 78.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 112.T + 9.40e3T^{2} \)
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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

−10.66446685898210091966430687992, −9.556731347976701766213895796279, −8.831132077072704407874201711090, −8.129425429424796761204217207840, −6.69083095538585588762847088957, −5.89217234514550403614343964077, −5.18612536046706502648949787513, −3.54664338165797659094602145812, −2.09706488989570714040400725556, −0.78258101752790498503656767067, 0.77798658495052976665593420504, 3.16566172996118700393786658903, 3.95931627557223396837567699814, 5.00084749813635171636286995034, 6.45166708284930817691737849279, 7.08943043252603179566222574219, 8.263834840485667593589448909875, 8.935890820324725397945130549925, 9.918782499603268584917969965386, 10.52141324793149680948422364228

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