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Properties

Label 2-525-21.5-c1-0-40
Degree $2$
Conductor $525$
Sign $-0.0224 + 0.999i$
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  + (0.780 + 0.450i)2-s + (−0.641 + 1.60i)3-s + (−0.593 − 1.02i)4-s + (−1.22 + 0.966i)6-s + (0.105 − 2.64i)7-s − 2.87i·8-s + (−2.17 − 2.06i)9-s + (−5.46 + 3.15i)11-s + (2.03 − 0.295i)12-s − 3.77i·13-s + (1.27 − 2.01i)14-s + (0.107 − 0.185i)16-s + (−1.88 − 3.27i)17-s + (−0.768 − 2.59i)18-s + (−4.57 − 2.64i)19-s + ⋯
L(s)  = 1  + (0.551 + 0.318i)2-s + (−0.370 + 0.928i)3-s + (−0.296 − 0.514i)4-s + (−0.500 + 0.394i)6-s + (0.0398 − 0.999i)7-s − 1.01i·8-s + (−0.725 − 0.688i)9-s + (−1.64 + 0.950i)11-s + (0.587 − 0.0853i)12-s − 1.04i·13-s + (0.340 − 0.538i)14-s + (0.0268 − 0.0464i)16-s + (−0.458 − 0.793i)17-s + (−0.181 − 0.611i)18-s + (−1.05 − 0.606i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0224 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0224 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0224 + 0.999i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.0224 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554837 - 0.567445i\)
\(L(\frac12)\) \(\approx\) \(0.554837 - 0.567445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.641 - 1.60i)T \)
5 \( 1 \)
7 \( 1 + (-0.105 + 2.64i)T \)
good2 \( 1 + (-0.780 - 0.450i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (5.46 - 3.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.77iT - 13T^{2} \)
17 \( 1 + (1.88 + 3.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.57 + 2.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.87 - 3.38i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.06iT - 29T^{2} \)
31 \( 1 + (-0.349 + 0.201i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.668 - 1.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.53T + 41T^{2} \)
43 \( 1 - 4.84T + 43T^{2} \)
47 \( 1 + (-0.970 + 1.68i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.24 - 0.720i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.60 - 2.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.3 + 6.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.55 + 2.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.21iT - 71T^{2} \)
73 \( 1 + (-2.18 + 1.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.05 - 5.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + (-0.590 + 1.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.10iT - 97T^{2} \)
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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

−10.58447517017792841369734510139, −9.964483660715338182559397632208, −9.140597850284801988159682476532, −7.76017766053316608886198694030, −6.85567313182544651029891385787, −5.65135459393934455042468615956, −4.87884508851754755536990234975, −4.31904435314589486914034018779, −2.91685374835163960585311154523, −0.38810096430815310231210820016, 2.14674022891055438743780856842, 2.95152094403355958678816781142, 4.53708914247654089683259609521, 5.51567840136960418635770972787, 6.28195955470393169136840517261, 7.57286135005466609768067029057, 8.482664898454624482808616962806, 8.862463869043453641914461851569, 10.75844209558348435662240494077, 11.14404639283626858987398082524

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