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Properties

Label 2-525-21.11-c0-0-0
Degree $2$
Conductor $525$
Sign $-0.832 - 0.553i$
Analytic cond. $0.262009$
Root an. cond. $0.511868$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s − 13-s + (−0.499 − 0.866i)16-s + (0.5 + 0.866i)19-s + (−0.499 − 0.866i)21-s + 0.999·27-s + (−0.499 − 0.866i)28-s + (−1 + 1.73i)31-s + 0.999·36-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s − 13-s + (−0.499 − 0.866i)16-s + (0.5 + 0.866i)19-s + (−0.499 − 0.866i)21-s + 0.999·27-s + (−0.499 − 0.866i)28-s + (−1 + 1.73i)31-s + 0.999·36-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(0.262009\)
Root analytic conductor: \(0.511868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :0),\ -0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5186970818\)
\(L(\frac12)\) \(\approx\) \(0.5186970818\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−11.57596629450047385294149256094, −10.44205479888571869153672961106, −9.523920392977881871352570832315, −9.036182472169267497533832636641, −8.005811688318986392368121371995, −6.86889865513247978688940839922, −5.65251499519200391774531028969, −4.87076031538828408721607969332, −3.73572400142241737225937913720, −2.79166038844617536012507684051, 0.68074450495621845550047861351, 2.34578355365979080511941240344, 4.16411433482244174297196362394, 5.23270637026177382040797364169, 6.09122957195238060392701248390, 7.08605525622351178061507309412, 7.72761338922032054464376005100, 9.158288463144063696394578367996, 9.833685930215084430878623295353, 10.81202333451768802602278382690

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