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Properties

Label 2-3528-392.157-c0-0-0
Degree $2$
Conductor $3528$
Sign $-0.518 - 0.855i$
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  + (0.149 − 0.988i)2-s + (−0.955 − 0.294i)4-s + (0.139 + 1.85i)5-s + (0.0747 − 0.997i)7-s + (−0.433 + 0.900i)8-s + (1.85 + 0.139i)10-s + (−1.77 + 0.698i)11-s + (−0.974 − 0.222i)14-s + (0.826 + 0.563i)16-s + (0.414 − 1.81i)20-s + (0.425 + 1.86i)22-s + (−2.43 + 0.367i)25-s + (−0.365 + 0.930i)28-s + (−1.92 − 0.440i)29-s + (−0.975 − 0.563i)31-s + (0.680 − 0.733i)32-s + ⋯
L(s)  = 1  + (0.149 − 0.988i)2-s + (−0.955 − 0.294i)4-s + (0.139 + 1.85i)5-s + (0.0747 − 0.997i)7-s + (−0.433 + 0.900i)8-s + (1.85 + 0.139i)10-s + (−1.77 + 0.698i)11-s + (−0.974 − 0.222i)14-s + (0.826 + 0.563i)16-s + (0.414 − 1.81i)20-s + (0.425 + 1.86i)22-s + (−2.43 + 0.367i)25-s + (−0.365 + 0.930i)28-s + (−1.92 − 0.440i)29-s + (−0.975 − 0.563i)31-s + (0.680 − 0.733i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.518 - 0.855i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.518 - 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2542142215\)
\(L(\frac12)\) \(\approx\) \(0.2542142215\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.149 + 0.988i)T \)
3 \( 1 \)
7 \( 1 + (-0.0747 + 0.997i)T \)
good5 \( 1 + (-0.139 - 1.85i)T + (-0.988 + 0.149i)T^{2} \)
11 \( 1 + (1.77 - 0.698i)T + (0.733 - 0.680i)T^{2} \)
13 \( 1 + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.0747 - 0.997i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (1.92 + 0.440i)T + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.975 + 0.563i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.826 + 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.955 - 0.294i)T^{2} \)
53 \( 1 + (0.294 - 0.955i)T + (-0.826 - 0.563i)T^{2} \)
59 \( 1 + (-0.129 + 1.72i)T + (-0.988 - 0.149i)T^{2} \)
61 \( 1 + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.129 - 0.858i)T + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + (-0.0747 - 0.129i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.185 + 0.233i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
97 \( 1 - 1.36iT - T^{2} \)
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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

−9.505869038369060722576742505667, −7.997828089640890309104878200732, −7.59742074330488430263262629956, −6.92333702996313489972262407014, −5.88303388192755699364581441744, −5.15198867767088241054688934357, −4.06158342074560937312193761089, −3.43498939230008464786215215949, −2.56028369187748176251327020772, −1.94006441462276588906695928363, 0.13083142315937017604533364868, 1.73519341401985450997068862956, 3.06332022004516452191112313378, 4.18225283451293434261287368973, 5.08714768083418865033223363225, 5.53328779741600579764815557340, 5.76788305609385951836155188537, 7.14401520181289029160080543110, 8.083013758237783328195670070931, 8.276720044334871573711164005852

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