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Properties

Label 2-525-21.5-c1-0-18
Degree $2$
Conductor $525$
Sign $-0.709 - 0.704i$
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  + (2.17 + 1.25i)2-s + (−0.831 + 1.51i)3-s + (2.14 + 3.71i)4-s + (−3.71 + 2.25i)6-s + (2.06 + 1.65i)7-s + 5.74i·8-s + (−1.61 − 2.52i)9-s + (1.48 − 0.859i)11-s + (−7.42 + 0.172i)12-s + 0.360i·13-s + (2.39 + 6.18i)14-s + (−2.91 + 5.04i)16-s + (−1.27 − 2.20i)17-s + (−0.348 − 7.51i)18-s + (−4.93 − 2.84i)19-s + ⋯
L(s)  = 1  + (1.53 + 0.886i)2-s + (−0.479 + 0.877i)3-s + (1.07 + 1.85i)4-s + (−1.51 + 0.922i)6-s + (0.778 + 0.627i)7-s + 2.03i·8-s + (−0.539 − 0.841i)9-s + (0.448 − 0.259i)11-s + (−2.14 + 0.0496i)12-s + 0.100i·13-s + (0.639 + 1.65i)14-s + (−0.727 + 1.26i)16-s + (−0.308 − 0.534i)17-s + (−0.0821 − 1.77i)18-s + (−1.13 − 0.653i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.709 - 0.704i$
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: \((2,\ 525,\ (\ :1/2),\ -0.709 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13804 + 2.76156i\)
\(L(\frac12)\) \(\approx\) \(1.13804 + 2.76156i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.831 - 1.51i)T \)
5 \( 1 \)
7 \( 1 + (-2.06 - 1.65i)T \)
good2 \( 1 + (-2.17 - 1.25i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-1.48 + 0.859i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.360iT - 13T^{2} \)
17 \( 1 + (1.27 + 2.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.93 + 2.84i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.17 - 1.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.76iT - 29T^{2} \)
31 \( 1 + (2.41 - 1.39i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.65 - 2.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.63T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.25 + 0.727i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.42 - 5.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.38 - 0.801i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.24 + 2.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.1iT - 71T^{2} \)
73 \( 1 + (10.0 - 5.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.93 - 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.50T + 83T^{2} \)
89 \( 1 + (-6.10 + 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.18iT - 97T^{2} \)
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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

−11.51197744439307618248812576990, −10.67494947073517044102498939824, −9.206850125637162142080346998824, −8.452587857712253314062156982136, −7.17585438522352650935845617857, −6.25629765459579526630386625052, −5.46914926368047045525561229210, −4.69165067783616132654667485269, −3.95779959430937249308983614939, −2.66164769333409367026542829608, 1.35380672975720922655877843456, 2.31135145891939414035156000584, 3.85856132762312130633592529747, 4.69538199887616943044609234015, 5.69226838172153953661017052155, 6.53585696327922042743236494975, 7.48562507526134124840922772452, 8.679593993322295625085519119428, 10.36591774550841112476677544643, 10.87322037039706571763881480332

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