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Properties

Label 2-525-5.4-c5-0-57
Degree $2$
Conductor $525$
Sign $0.447 + 0.894i$
Analytic cond. $84.2015$
Root an. cond. $9.17613$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 9i·3-s + 31·4-s − 9·6-s + 49i·7-s − 63i·8-s − 81·9-s − 340·11-s − 279i·12-s + 454i·13-s + 49·14-s + 929·16-s + 798i·17-s + 81i·18-s − 892·19-s + ⋯
L(s)  = 1  − 0.176i·2-s − 0.577i·3-s + 0.968·4-s − 0.102·6-s + 0.377i·7-s − 0.348i·8-s − 0.333·9-s − 0.847·11-s − 0.559i·12-s + 0.745i·13-s + 0.0668·14-s + 0.907·16-s + 0.669i·17-s + 0.0589i·18-s − 0.566·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(84.2015\)
Root analytic conductor: \(9.17613\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :5/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.578964182\)
\(L(\frac12)\) \(\approx\) \(2.578964182\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9iT \)
5 \( 1 \)
7 \( 1 - 49iT \)
good2 \( 1 + iT - 32T^{2} \)
11 \( 1 + 340T + 1.61e5T^{2} \)
13 \( 1 - 454iT - 3.71e5T^{2} \)
17 \( 1 - 798iT - 1.41e6T^{2} \)
19 \( 1 + 892T + 2.47e6T^{2} \)
23 \( 1 + 3.19e3iT - 6.43e6T^{2} \)
29 \( 1 - 8.24e3T + 2.05e7T^{2} \)
31 \( 1 + 2.49e3T + 2.86e7T^{2} \)
37 \( 1 + 9.79e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.98e4T + 1.15e8T^{2} \)
43 \( 1 + 1.72e4iT - 1.47e8T^{2} \)
47 \( 1 + 8.92e3iT - 2.29e8T^{2} \)
53 \( 1 - 150iT - 4.18e8T^{2} \)
59 \( 1 - 4.23e4T + 7.14e8T^{2} \)
61 \( 1 - 1.47e4T + 8.44e8T^{2} \)
67 \( 1 - 1.67e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.45e4T + 1.80e9T^{2} \)
73 \( 1 - 7.83e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.27e3T + 3.07e9T^{2} \)
83 \( 1 + 3.77e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.17e5T + 5.58e9T^{2} \)
97 \( 1 + 1.00e4iT - 8.58e9T^{2} \)
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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

−10.21877312442153472369271278421, −8.841570237401453904916397195269, −8.093359423733256485403281274681, −7.07192354727656082623212264657, −6.38965058811131032132936348056, −5.47555024839076003730230663124, −4.05388294609623167481252015840, −2.60346247328115620667975010029, −2.07527145332434108218637031657, −0.66932771822843891191417486954, 0.937123339118542102148558613541, 2.49337970870526630662315142631, 3.28352992719996725955775250561, 4.68372193317830467201781981675, 5.60513849393895086389154288704, 6.55510443540717000154773896784, 7.62301333734901828672932142925, 8.201232510173380377435456814628, 9.539930802938821547542978563384, 10.34855154126651163910116529711

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