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Properties

Label 2-525-15.8-c1-0-30
Degree $2$
Conductor $525$
Sign $-0.736 + 0.676i$
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  + (−0.723 − 0.723i)2-s + (1.41 − 0.994i)3-s − 0.951i·4-s + (−1.74 − 0.307i)6-s + (0.707 − 0.707i)7-s + (−2.13 + 2.13i)8-s + (1.02 − 2.81i)9-s − 3.63i·11-s + (−0.946 − 1.34i)12-s + (4.19 + 4.19i)13-s − 1.02·14-s + 1.19·16-s + (−4.61 − 4.61i)17-s + (−2.78 + 1.30i)18-s − 3.77i·19-s + ⋯
L(s)  = 1  + (−0.511 − 0.511i)2-s + (0.818 − 0.573i)3-s − 0.475i·4-s + (−0.713 − 0.125i)6-s + (0.267 − 0.267i)7-s + (−0.755 + 0.755i)8-s + (0.341 − 0.939i)9-s − 1.09i·11-s + (−0.273 − 0.389i)12-s + (1.16 + 1.16i)13-s − 0.273·14-s + 0.297·16-s + (−1.11 − 1.11i)17-s + (−0.655 + 0.306i)18-s − 0.865i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.736 + 0.676i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.501473 - 1.28832i\)
\(L(\frac12)\) \(\approx\) \(0.501473 - 1.28832i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.41 + 0.994i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (0.723 + 0.723i)T + 2iT^{2} \)
11 \( 1 + 3.63iT - 11T^{2} \)
13 \( 1 + (-4.19 - 4.19i)T + 13iT^{2} \)
17 \( 1 + (4.61 + 4.61i)T + 17iT^{2} \)
19 \( 1 + 3.77iT - 19T^{2} \)
23 \( 1 + (1.81 - 1.81i)T - 23iT^{2} \)
29 \( 1 - 1.13T + 29T^{2} \)
31 \( 1 - 3.62T + 31T^{2} \)
37 \( 1 + (7.24 - 7.24i)T - 37iT^{2} \)
41 \( 1 - 0.314iT - 41T^{2} \)
43 \( 1 + (-1.06 - 1.06i)T + 43iT^{2} \)
47 \( 1 + (-4.48 - 4.48i)T + 47iT^{2} \)
53 \( 1 + (-1.44 + 1.44i)T - 53iT^{2} \)
59 \( 1 + 8.70T + 59T^{2} \)
61 \( 1 - 3.08T + 61T^{2} \)
67 \( 1 + (-9.67 + 9.67i)T - 67iT^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (0.710 + 0.710i)T + 73iT^{2} \)
79 \( 1 + 7.30iT - 79T^{2} \)
83 \( 1 + (-9.58 + 9.58i)T - 83iT^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + (5.28 - 5.28i)T - 97iT^{2} \)
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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

−10.58361134460880452691634094634, −9.334375751220943638913566265854, −8.926680522642669972233215494740, −8.176606610668107774677940113394, −6.84295676280801697010879347552, −6.18686043895751553626496456360, −4.68720733013776926443130660406, −3.30143783232213113336086755330, −2.12711571825249717476688380830, −0.913146028770952390182780595199, 2.11990239061543821824530320367, 3.51096918911062366296162544066, 4.30306516145502655642459620606, 5.75230877493570599887585008123, 6.90544739117300794983101198010, 7.984923836158526122447612493816, 8.406849561188103534279170214614, 9.153836110156798756374250303533, 10.20255568671312124829328125068, 10.81167268012665741857491439921

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