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Properties

Label 2-525-7.2-c1-0-8
Degree $2$
Conductor $525$
Sign $0.922 - 0.386i$
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  + (0.366 − 0.633i)2-s + (−0.5 − 0.866i)3-s + (0.732 + 1.26i)4-s − 0.732·6-s + (0.866 + 2.5i)7-s + 2.53·8-s + (−0.499 + 0.866i)9-s + (1.36 + 2.36i)11-s + (0.732 − 1.26i)12-s − 5.73·13-s + (1.90 + 0.366i)14-s + (−0.535 + 0.928i)16-s + (3.36 + 5.83i)17-s + (0.366 + 0.633i)18-s + (1.23 − 2.13i)19-s + ⋯
L(s)  = 1  + (0.258 − 0.448i)2-s + (−0.288 − 0.499i)3-s + (0.366 + 0.633i)4-s − 0.298·6-s + (0.327 + 0.944i)7-s + 0.896·8-s + (−0.166 + 0.288i)9-s + (0.411 + 0.713i)11-s + (0.211 − 0.366i)12-s − 1.58·13-s + (0.508 + 0.0978i)14-s + (−0.133 + 0.232i)16-s + (0.816 + 1.41i)17-s + (0.0862 + 0.149i)18-s + (0.282 − 0.489i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.922 - 0.386i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.922 - 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64051 + 0.329858i\)
\(L(\frac12)\) \(\approx\) \(1.64051 + 0.329858i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-0.866 - 2.5i)T \)
good2 \( 1 + (-0.366 + 0.633i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-1.36 - 2.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.73T + 13T^{2} \)
17 \( 1 + (-3.36 - 5.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.23 + 2.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.633 - 1.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 + (3.23 + 5.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.59 + 6.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.73T + 41T^{2} \)
43 \( 1 - 7.19T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.19 + 7.26i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.09 + 8.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.33 - 2.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.19T + 71T^{2} \)
73 \( 1 + (2.33 + 4.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.69 - 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 + (-4.56 + 7.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.07T + 97T^{2} \)
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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

−11.21982464443725909403993618401, −10.17327762510206811959716869625, −9.189955612558031819505390221943, −7.961817048694697484778796129073, −7.45816648713821278374193928382, −6.35127040971011091124678852270, −5.22959840953971991922015199959, −4.16772047827140247390038108307, −2.69733941925457171605586988936, −1.84943518277618743964910931655, 1.00669707170838769025332465468, 2.93910779492461082331952835779, 4.47691523437194824026953194871, 5.09085576384468473926374409261, 6.12766302616740964396092817931, 7.15963689908187524841796891918, 7.76051782627422204414350666907, 9.283573456251249277876300656953, 10.06736278913121675714496319011, 10.65959356282396108341195310348

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