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Properties

Label 2-3528-504.227-c0-0-2
Degree $2$
Conductor $3528$
Sign $0.585 - 0.810i$
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  + i·2-s + (−0.608 + 0.793i)3-s − 4-s + (−0.793 − 0.608i)6-s − i·8-s + (−0.258 − 0.965i)9-s + (−1.67 + 0.965i)11-s + (0.608 − 0.793i)12-s + 16-s + (0.793 − 1.37i)17-s + (0.965 − 0.258i)18-s + (1.71 − 0.991i)19-s + (−0.965 − 1.67i)22-s + (0.793 + 0.608i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + i·2-s + (−0.608 + 0.793i)3-s − 4-s + (−0.793 − 0.608i)6-s − i·8-s + (−0.258 − 0.965i)9-s + (−1.67 + 0.965i)11-s + (0.608 − 0.793i)12-s + 16-s + (0.793 − 1.37i)17-s + (0.965 − 0.258i)18-s + (1.71 − 0.991i)19-s + (−0.965 − 1.67i)22-s + (0.793 + 0.608i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.585 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7261995746\)
\(L(\frac12)\) \(\approx\) \(0.7261995746\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.793 + 1.37i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.71 + 0.991i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.608 + 1.05i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 - 0.261T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.73T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.226 - 0.130i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.05 - 0.608i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.916307283603707775635429316146, −7.87742803586172568147358731989, −7.38936402286969688146887835026, −6.69112085114995514574668895662, −5.60738697578161643822310418818, −5.07855289284837099266407302610, −4.78471898175022798360152041620, −3.59044527470356049877362531058, −2.68685455379265789290548670752, −0.58734762036789684022756430096, 1.01297716075002407220536131696, 1.94504590333269035776144020908, 3.04774075330184909668908027924, 3.66254057055812569169334117319, 5.11410952077518522781201815578, 5.47200865476701208607243933449, 6.09858229614069490080124990210, 7.44266079556237434826298113287, 8.051599576238795340266343417419, 8.365158501916408312641182426494

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