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Properties

Label 2-2872-2872.717-c0-0-2
Degree $2$
Conductor $2872$
Sign $-0.986 + 0.164i$
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  + (−0.546 − 0.837i)2-s + 1.93i·3-s + (−0.401 + 0.915i)4-s + 1.99i·5-s + (1.62 − 1.06i)6-s + (0.986 − 0.164i)8-s − 2.75·9-s + (1.66 − 1.09i)10-s + 1.22i·11-s + (−1.77 − 0.778i)12-s − 3.86·15-s + (−0.677 − 0.735i)16-s + 1.35·17-s + (1.50 + 2.30i)18-s + (−1.82 − 0.800i)20-s + ⋯
L(s)  = 1  + (−0.546 − 0.837i)2-s + 1.93i·3-s + (−0.401 + 0.915i)4-s + 1.99i·5-s + (1.62 − 1.06i)6-s + (0.986 − 0.164i)8-s − 2.75·9-s + (1.66 − 1.09i)10-s + 1.22i·11-s + (−1.77 − 0.778i)12-s − 3.86·15-s + (−0.677 − 0.735i)16-s + 1.35·17-s + (1.50 + 2.30i)18-s + (−1.82 − 0.800i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2872\)    =    \(2^{3} \cdot 359\)
Sign: $-0.986 + 0.164i$
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: \((2,\ 2872,\ (\ :0),\ -0.986 + 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7856218416\)
\(L(\frac12)\) \(\approx\) \(0.7856218416\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.546 + 0.837i)T \)
359 \( 1 + T \)
good3 \( 1 - 1.93iT - T^{2} \)
5 \( 1 - 1.99iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.22iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 1.35T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 0.803T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.649iT - T^{2} \)
41 \( 1 - 1.89T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.75T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.09T + T^{2} \)
79 \( 1 - 0.165T + T^{2} \)
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

−9.801713776083383442778785824340, −9.039402078923596944761513718481, −7.88697790970117633242176203627, −7.40378962676641564189608334734, −6.23587487552907385031388872016, −5.33359209871276042139587235884, −4.15637864088411236725842103814, −3.80117252747448064583199042420, −2.89363184291206467085385489010, −2.34891349665473470719827473796, 0.70622560025376278101023908869, 1.13800666926895219897710244748, 2.24873905499384032082559976584, 3.91380092434425878679202301547, 5.24320692851039369281427157890, 5.79655412386246138485407838076, 6.09767631446252617883313192729, 7.41132551515169116485681338536, 7.80176944852138844125937946742, 8.394770098834052542121572204985

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