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Properties

Label 2-2872-2872.717-c0-0-18
Degree $2$
Conductor $2872$
Sign $-0.0825 + 0.996i$
Analytic cond. $1.43331$
Root an. cond. $1.19721$
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  + (0.879 − 0.475i)2-s − 1.22i·3-s + (0.546 − 0.837i)4-s + 1.47i·5-s + (−0.584 − 1.08i)6-s + (0.0825 − 0.996i)8-s − 0.509·9-s + (0.700 + 1.29i)10-s − 0.649i·11-s + (−1.02 − 0.671i)12-s + 1.80·15-s + (−0.401 − 0.915i)16-s + 0.803·17-s + (−0.447 + 0.242i)18-s + (1.23 + 0.804i)20-s + ⋯
L(s)  = 1  + (0.879 − 0.475i)2-s − 1.22i·3-s + (0.546 − 0.837i)4-s + 1.47i·5-s + (−0.584 − 1.08i)6-s + (0.0825 − 0.996i)8-s − 0.509·9-s + (0.700 + 1.29i)10-s − 0.649i·11-s + (−1.02 − 0.671i)12-s + 1.80·15-s + (−0.401 − 0.915i)16-s + 0.803·17-s + (−0.447 + 0.242i)18-s + (1.23 + 0.804i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0825 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0825 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2872\)    =    \(2^{3} \cdot 359\)
Sign: $-0.0825 + 0.996i$
Analytic conductor: \(1.43331\)
Root analytic conductor: \(1.19721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2872} (717, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2872,\ (\ :0),\ -0.0825 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.168678222\)
\(L(\frac12)\) \(\approx\) \(2.168678222\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.879 + 0.475i)T \)
359 \( 1 + T \)
good3 \( 1 + 1.22iT - T^{2} \)
5 \( 1 - 1.47iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 0.649iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 0.803T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.09T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.329iT - T^{2} \)
41 \( 1 + 1.97T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 0.490T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.75T + T^{2} \)
79 \( 1 - 1.35T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
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

−8.621864500287685893411307632862, −7.59843812744964636691718967438, −7.06774493645176571968259457309, −6.49512491883702337907783376066, −5.89886607685548456253535262736, −4.95550744244193233731780972792, −3.61542676638207263246081089395, −3.04840675810374915613835132391, −2.23878208071234479440499139984, −1.17158710316032038132105881279, 1.60258878383849255843666619005, 3.07514735400921246083484766653, 3.92577969405626925969424297619, 4.62481436734794452388020874698, 5.10654041730615530454617274428, 5.64184908076251789718697988599, 6.83004858293188781878495641321, 7.63201051655516793655255150583, 8.539801380879298333592738775352, 9.018908027348009740411786789853

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