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Properties

Label 2-3528-504.499-c0-0-8
Degree $2$
Conductor $3528$
Sign $-0.458 + 0.888i$
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 + 3-s − 4-s + (0.866 − 0.5i)5-s − i·6-s + i·8-s + 9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s − 12-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)15-s + 16-s + (−0.5 − 0.866i)17-s − i·18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  − i·2-s + 3-s − 4-s + (0.866 − 0.5i)5-s − i·6-s + i·8-s + 9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s − 12-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)15-s + 16-s + (−0.5 − 0.866i)17-s − i·18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.458 + 0.888i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.877128719\)
\(L(\frac12)\) \(\approx\) \(1.877128719\)
\(L(1)\) 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 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.850108887785925816568770717487, −7.997289768555565186344613845434, −7.29551751361889096626349409456, −6.11728790104566088773916390454, −5.20070930955735854441343998237, −4.59278107555452153253063369024, −3.54587914931911426509636225280, −2.82480991154033443372409468280, −2.05318900107679578454500408818, −1.03841722139716003820846220281, 1.76290038859007705313683713133, 2.43918535839094800802980463391, 3.89974711375387114705896124915, 4.20557831668578490276161422937, 5.37650265246946683570207605294, 6.18874053534362900128818851912, 6.86600826910532121848170777621, 7.46356525309218799940392762534, 8.147686135497168060419067504303, 8.963219561580619700377926210688

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