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Properties

Label 2-525-25.21-c1-0-8
Degree $2$
Conductor $525$
Sign $0.637 - 0.770i$
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  + (0.190 − 0.587i)2-s + (0.809 + 0.587i)3-s + (1.30 + 0.951i)4-s − 2.23·5-s + (0.5 − 0.363i)6-s − 7-s + (1.80 − 1.31i)8-s + (0.309 + 0.951i)9-s + (−0.427 + 1.31i)10-s + (−1.61 + 4.97i)11-s + (0.5 + 1.53i)12-s + (1.69 + 5.20i)13-s + (−0.190 + 0.587i)14-s + (−1.80 − 1.31i)15-s + (0.572 + 1.76i)16-s + (6.04 − 4.39i)17-s + ⋯
L(s)  = 1  + (0.135 − 0.415i)2-s + (0.467 + 0.339i)3-s + (0.654 + 0.475i)4-s − 0.999·5-s + (0.204 − 0.148i)6-s − 0.377·7-s + (0.639 − 0.464i)8-s + (0.103 + 0.317i)9-s + (−0.135 + 0.415i)10-s + (−0.487 + 1.50i)11-s + (0.144 + 0.444i)12-s + (0.468 + 1.44i)13-s + (−0.0510 + 0.157i)14-s + (−0.467 − 0.339i)15-s + (0.143 + 0.440i)16-s + (1.46 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57049 + 0.739020i\)
\(L(\frac12)\) \(\approx\) \(1.57049 + 0.739020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + 2.23T \)
7 \( 1 + T \)
good2 \( 1 + (-0.190 + 0.587i)T + (-1.61 - 1.17i)T^{2} \)
11 \( 1 + (1.61 - 4.97i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.69 - 5.20i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-6.04 + 4.39i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.11 + 0.812i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.809 - 2.48i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (5.42 + 3.94i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.30 - 1.67i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.42 - 7.46i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.88 + 5.79i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (9.70 + 7.05i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-5.73 - 4.16i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.64 + 5.06i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.23 + 13.0i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.28 + 6.01i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-6.04 - 4.39i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.899 + 2.76i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.1 + 7.38i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4 - 2.90i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.38 - 4.25i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (2.04 + 1.48i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.22233285394256636853970942231, −10.02316019620915841787307390990, −9.449998197785439826612566739999, −8.135489597172423695788798176984, −7.38469421984987685418205477244, −6.82178126111040023748214800129, −5.00741716684383703557451421431, −3.97734129503180064768784414404, −3.22043251743417688051139364853, −1.93213126305064007059107443207, 0.984431225127897406254179517371, 2.95321747653964698216294685346, 3.66494612750793932267745417923, 5.51621562978604837944520528828, 5.99838225634515335205206757929, 7.28880710560473519556560070841, 7.994036337781449379638609817912, 8.494120679181085654784928271219, 10.03924909211592425895635872985, 10.80419395759371969079012160841

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