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Properties

Label 2-525-35.33-c1-0-1
Degree $2$
Conductor $525$
Sign $-0.981 - 0.192i$
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  + (0.461 + 1.72i)2-s + (−0.965 − 0.258i)3-s + (−1.01 + 0.587i)4-s − 1.78i·6-s + (−2.04 + 1.68i)7-s + (1.03 + 1.03i)8-s + (0.866 + 0.499i)9-s + (1.46 + 2.54i)11-s + (1.13 − 0.304i)12-s + (−0.187 + 0.187i)13-s + (−3.83 − 2.74i)14-s + (−2.48 + 4.30i)16-s + (0.868 − 3.24i)17-s + (−0.461 + 1.72i)18-s + (−1.81 + 3.14i)19-s + ⋯
L(s)  = 1  + (0.326 + 1.21i)2-s + (−0.557 − 0.149i)3-s + (−0.508 + 0.293i)4-s − 0.727i·6-s + (−0.772 + 0.635i)7-s + (0.367 + 0.367i)8-s + (0.288 + 0.166i)9-s + (0.442 + 0.766i)11-s + (0.327 − 0.0877i)12-s + (−0.0521 + 0.0521i)13-s + (−1.02 − 0.732i)14-s + (−0.621 + 1.07i)16-s + (0.210 − 0.786i)17-s + (−0.108 + 0.405i)18-s + (−0.417 + 0.722i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.981 - 0.192i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.981 - 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111421 + 1.14614i\)
\(L(\frac12)\) \(\approx\) \(0.111421 + 1.14614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (2.04 - 1.68i)T \)
good2 \( 1 + (-0.461 - 1.72i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-1.46 - 2.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.187 - 0.187i)T - 13iT^{2} \)
17 \( 1 + (-0.868 + 3.24i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.81 - 3.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (9.07 - 2.43i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.815iT - 29T^{2} \)
31 \( 1 + (3.76 - 2.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.69 - 6.31i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.45iT - 41T^{2} \)
43 \( 1 + (-3.59 - 3.59i)T + 43iT^{2} \)
47 \( 1 + (-9.24 + 2.47i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.30 + 4.87i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.41 + 9.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.07 - 4.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-15.3 - 4.10i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.68T + 71T^{2} \)
73 \( 1 + (1.89 + 0.508i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.44 - 2.56i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.09 + 6.09i)T - 83iT^{2} \)
89 \( 1 + (-4.87 + 8.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.93 + 5.93i)T + 97iT^{2} \)
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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

−11.48245867463665506106630654159, −10.21889017854520792678838779293, −9.494442329322687832098505837443, −8.306273186007896456307514882659, −7.39682540687022201323056541039, −6.56600294772771465942012113826, −5.91746735955896587059032338346, −5.06913917548104361920674060875, −3.90128732061339758520769090288, −2.06083004763963988791581343080, 0.63990619983288836337120284428, 2.29818894819094489735506171935, 3.74161767688119051302141247779, 4.15633438660542824212671098510, 5.73978248195580254828251553965, 6.60125864600014340531001946994, 7.67712260283904847751893010523, 9.058186058373312044329056634077, 9.954483625170286076759949748977, 10.68470946988189554413494345350

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