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Properties

Label 2-3528-504.59-c0-0-6
Degree $2$
Conductor $3528$
Sign $0.252 - 0.967i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.923 + 0.382i)3-s + (0.499 + 0.866i)4-s + (0.608 + 0.793i)6-s + 0.999i·8-s + (0.707 + 0.707i)9-s − 1.93i·11-s + (0.130 + 0.991i)12-s + (−0.5 + 0.866i)16-s + (−0.608 + 1.05i)17-s + (0.258 + 0.965i)18-s + (0.226 − 0.130i)19-s + (0.965 − 1.67i)22-s + (−0.382 + 0.923i)24-s + 25-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.923 + 0.382i)3-s + (0.499 + 0.866i)4-s + (0.608 + 0.793i)6-s + 0.999i·8-s + (0.707 + 0.707i)9-s − 1.93i·11-s + (0.130 + 0.991i)12-s + (−0.5 + 0.866i)16-s + (−0.608 + 1.05i)17-s + (0.258 + 0.965i)18-s + (0.226 − 0.130i)19-s + (0.965 − 1.67i)22-s + (−0.382 + 0.923i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.943754620\)
\(L(\frac12)\) \(\approx\) \(2.943754620\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 + 1.93iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.608 - 1.05i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.226 + 0.130i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.793 - 1.37i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.991 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.71 + 0.991i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.37 + 0.793i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.595448459157038270236395244828, −8.267254852047261200773080838218, −7.50517109260331630735798694711, −6.49453142418192702341773797433, −5.98933357643624937568636871414, −4.97566436305939204507404408293, −4.30412944085350944317706930736, −3.24726948258904171904850867053, −3.08016006201003142351612666612, −1.72765604056117275265681610550, 1.35801474960080208525360684544, 2.28442108750984313071904956363, 2.86296785992208431066974183476, 3.97885230913438380838124793042, 4.57073980349798896780162753233, 5.33456597748952811770627332836, 6.52528982006188074407393846067, 7.18742826293026261826814738709, 7.46585792752944296771464436958, 8.845435967034822112317084380395

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