Properties

Label 2-525-35.27-c1-0-11
Degree $2$
Conductor $525$
Sign $0.967 - 0.252i$
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  + (0.167 + 0.167i)2-s + (0.707 + 0.707i)3-s − 1.94i·4-s + 0.236i·6-s + (0.0627 + 2.64i)7-s + (0.658 − 0.658i)8-s + 1.00i·9-s + 3.98·11-s + (1.37 − 1.37i)12-s + (0.500 + 0.500i)13-s + (−0.431 + 0.452i)14-s − 3.66·16-s + (−1.67 + 1.67i)17-s + (−0.167 + 0.167i)18-s + 7.21·19-s + ⋯
L(s)  = 1  + (0.118 + 0.118i)2-s + (0.408 + 0.408i)3-s − 0.972i·4-s + 0.0964i·6-s + (0.0237 + 0.999i)7-s + (0.232 − 0.232i)8-s + 0.333i·9-s + 1.20·11-s + (0.396 − 0.396i)12-s + (0.138 + 0.138i)13-s + (−0.115 + 0.120i)14-s − 0.917·16-s + (−0.407 + 0.407i)17-s + (−0.0393 + 0.0393i)18-s + 1.65·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.967 - 0.252i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86923 + 0.240131i\)
\(L(\frac12)\) \(\approx\) \(1.86923 + 0.240131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-0.0627 - 2.64i)T \)
good2 \( 1 + (-0.167 - 0.167i)T + 2iT^{2} \)
11 \( 1 - 3.98T + 11T^{2} \)
13 \( 1 + (-0.500 - 0.500i)T + 13iT^{2} \)
17 \( 1 + (1.67 - 1.67i)T - 17iT^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
23 \( 1 + (-5.16 + 5.16i)T - 23iT^{2} \)
29 \( 1 + 3.65iT - 29T^{2} \)
31 \( 1 - 4.93iT - 31T^{2} \)
37 \( 1 + (0.292 + 0.292i)T + 37iT^{2} \)
41 \( 1 + 7.63iT - 41T^{2} \)
43 \( 1 + (3.65 - 3.65i)T - 43iT^{2} \)
47 \( 1 + (0.305 - 0.305i)T - 47iT^{2} \)
53 \( 1 + (5.39 - 5.39i)T - 53iT^{2} \)
59 \( 1 + 6.10T + 59T^{2} \)
61 \( 1 - 7.11iT - 61T^{2} \)
67 \( 1 + (0.944 + 0.944i)T + 67iT^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + (-1.38 - 1.38i)T + 73iT^{2} \)
79 \( 1 - 8.64iT - 79T^{2} \)
83 \( 1 + (11.9 + 11.9i)T + 83iT^{2} \)
89 \( 1 + 7.82T + 89T^{2} \)
97 \( 1 + (-7.43 + 7.43i)T - 97iT^{2} \)
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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

−10.87508678106785940374935654669, −9.828528760773129533212350426182, −9.168436506232664929318320492965, −8.573952846970246456565201277805, −7.09659900256122732185745173777, −6.19593834325703477339399965111, −5.28232865616807531736055965753, −4.33063957838689459831749959797, −2.93115395716072891321379712276, −1.49109360051312326490333397566, 1.34855093321942638468764334372, 3.10577318938503888844455859723, 3.77864402965104487677564410719, 4.95997484460317782175153826673, 6.61474350320082540964741198712, 7.28448728136676736440392404738, 7.959473958907590513651388044189, 9.076592343278929585738829195322, 9.700614381267880972299920535754, 11.21344994965257865154651060580

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