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Properties

Label 2-525-35.13-c1-0-8
Degree $2$
Conductor $525$
Sign $0.320 - 0.947i$
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  + (−1.11 + 1.11i)2-s + (0.707 − 0.707i)3-s − 0.476i·4-s + 1.57i·6-s + (2.09 + 1.62i)7-s + (−1.69 − 1.69i)8-s − 1.00i·9-s + 5.20·11-s + (−0.336 − 0.336i)12-s + (−2.22 + 2.22i)13-s + (−4.13 + 0.523i)14-s + 4.72·16-s + (−3.11 − 3.11i)17-s + (1.11 + 1.11i)18-s + 4.13·19-s + ⋯
L(s)  = 1  + (−0.786 + 0.786i)2-s + (0.408 − 0.408i)3-s − 0.238i·4-s + 0.642i·6-s + (0.790 + 0.612i)7-s + (−0.599 − 0.599i)8-s − 0.333i·9-s + 1.56·11-s + (−0.0971 − 0.0971i)12-s + (−0.618 + 0.618i)13-s + (−1.10 + 0.140i)14-s + 1.18·16-s + (−0.755 − 0.755i)17-s + (0.262 + 0.262i)18-s + 0.947·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.320 - 0.947i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.983248 + 0.704993i\)
\(L(\frac12)\) \(\approx\) \(0.983248 + 0.704993i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-2.09 - 1.62i)T \)
good2 \( 1 + (1.11 - 1.11i)T - 2iT^{2} \)
11 \( 1 - 5.20T + 11T^{2} \)
13 \( 1 + (2.22 - 2.22i)T - 13iT^{2} \)
17 \( 1 + (3.11 + 3.11i)T + 17iT^{2} \)
19 \( 1 - 4.13T + 19T^{2} \)
23 \( 1 + (-2.97 - 2.97i)T + 23iT^{2} \)
29 \( 1 + 0.249iT - 29T^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 + (-1.69 + 1.69i)T - 37iT^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + (-4.39 - 4.39i)T + 43iT^{2} \)
47 \( 1 + (-3.11 - 3.11i)T + 47iT^{2} \)
53 \( 1 + (-4.89 - 4.89i)T + 53iT^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 + 2.16iT - 61T^{2} \)
67 \( 1 + (-2.50 + 2.50i)T - 67iT^{2} \)
71 \( 1 + 0.798T + 71T^{2} \)
73 \( 1 + (5.94 - 5.94i)T - 73iT^{2} \)
79 \( 1 + 5.29iT - 79T^{2} \)
83 \( 1 + (4.59 - 4.59i)T - 83iT^{2} \)
89 \( 1 + 6.29T + 89T^{2} \)
97 \( 1 + (12.1 + 12.1i)T + 97iT^{2} \)
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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

−11.13034515898750845247350512939, −9.481966371556558123763758490455, −9.147140905540575251525464853275, −8.461399665806072468566876096672, −7.27748195608295763603231171734, −6.97135353713870660262813247036, −5.77059914738187572126369201620, −4.42904155369453995336619350032, −3.00684676848328385601097420671, −1.41378532866919579863678073274, 1.07173290501172349809524235793, 2.34428709057701713126289858742, 3.73318632693921028709374175722, 4.75994475600494159437269231347, 6.06404729235560344256938800739, 7.37810095222974259077733037648, 8.325655855347026616277510684847, 9.111285411152588645979575575940, 9.808218169397922145312332100041, 10.60735379177043855438668714049

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