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Properties

Label 2-525-35.9-c1-0-4
Degree $2$
Conductor $525$
Sign $0.999 - 0.0370i$
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  + (−1.06 − 0.614i)2-s + (0.866 − 0.5i)3-s + (−0.245 − 0.424i)4-s − 1.22·6-s + (2.33 + 1.24i)7-s + 3.05i·8-s + (0.499 − 0.866i)9-s + (2.37 + 4.11i)11-s + (−0.424 − 0.245i)12-s + 6.25i·13-s + (−1.71 − 2.75i)14-s + (1.38 − 2.40i)16-s + (−5.44 + 3.14i)17-s + (−1.06 + 0.614i)18-s + (2.47 − 4.28i)19-s + ⋯
L(s)  = 1  + (−0.752 − 0.434i)2-s + (0.499 − 0.288i)3-s + (−0.122 − 0.212i)4-s − 0.501·6-s + (0.882 + 0.470i)7-s + 1.08i·8-s + (0.166 − 0.288i)9-s + (0.715 + 1.23i)11-s + (−0.122 − 0.0707i)12-s + 1.73i·13-s + (−0.459 − 0.737i)14-s + (0.347 − 0.601i)16-s + (−1.32 + 0.762i)17-s + (−0.250 + 0.144i)18-s + (0.567 − 0.983i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.999 - 0.0370i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.999 - 0.0370i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15030 + 0.0212926i\)
\(L(\frac12)\) \(\approx\) \(1.15030 + 0.0212926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.33 - 1.24i)T \)
good2 \( 1 + (1.06 + 0.614i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-2.37 - 4.11i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.25iT - 13T^{2} \)
17 \( 1 + (5.44 - 3.14i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.47 + 4.28i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.31 - 2.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (0.829 + 1.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.84 - 2.21i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.30T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + (3.26 + 1.88i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.23 - 3.60i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.117 - 0.203i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.15 + 1.24i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.06T + 71T^{2} \)
73 \( 1 + (-10.8 + 6.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.415 - 0.719i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.14iT - 83T^{2} \)
89 \( 1 + (-1.14 + 1.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.476iT - 97T^{2} \)
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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

−11.05019698934098347841356025558, −9.572472001080927335038692377812, −9.204208631725416805987850775890, −8.560580514642879324566460445682, −7.38310671190833718561035004724, −6.54873445344916394981784636908, −5.02933117453639199041027035319, −4.21515894868587830499544118573, −2.19962865207576942532167656629, −1.63580945855839485378966123008, 0.914743042091760605761196334855, 3.03318099925028175991731550645, 3.99854133807605423114347100393, 5.21262895748053185941045523191, 6.52730370802484989092317633462, 7.66975977535284089886714577190, 8.154809501280346941089973917463, 8.919043562882680538070047076500, 9.718750482813327463367728506760, 10.77912960636588934919609512810

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