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Properties

Label 2-3528-56.5-c0-0-2
Degree $2$
Conductor $3528$
Sign $0.832 - 0.553i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.999·8-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)23-s + (0.5 + 0.866i)25-s + (−0.499 − 0.866i)32-s + (0.999 + 1.73i)46-s − 0.999·50-s + 0.999·64-s + 2·71-s + (1 − 1.73i)79-s − 1.99·92-s + (0.499 − 0.866i)100-s + 2·113-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.999·8-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)23-s + (0.5 + 0.866i)25-s + (−0.499 − 0.866i)32-s + (0.999 + 1.73i)46-s − 0.999·50-s + 0.999·64-s + 2·71-s + (1 − 1.73i)79-s − 1.99·92-s + (0.499 − 0.866i)100-s + 2·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9418527445\)
\(L(\frac12)\) \(\approx\) \(0.9418527445\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.764276534352796418093402988094, −8.113849675588939971724448457477, −7.28062807232635100442161105862, −6.70290956732704815442400434377, −6.00103429938313683834301076969, −5.07435534125960691222377255127, −4.54254491198348336824201808673, −3.38521701402034353692838953883, −2.16773559856276974588629230442, −0.872533979805025585571220951764, 1.01558521462426583191447861259, 2.10769102970556075870322025558, 3.07061587324045367411451498667, 3.80087719155684335811447831703, 4.74870015289386919784856328243, 5.49372472663164425704655747751, 6.65922305597647123651756322668, 7.37033348274180819908720764979, 8.112070504998102084730376523797, 8.777602922116712621764802125922

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