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Properties

Label 2-2872-2872.717-c0-0-1
Degree $2$
Conductor $2872$
Sign $-0.677 + 0.735i$
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  + (−0.245 + 0.969i)2-s + 0.649i·3-s + (−0.879 − 0.475i)4-s + 1.83i·5-s + (−0.629 − 0.159i)6-s + (0.677 − 0.735i)8-s + 0.578·9-s + (−1.77 − 0.449i)10-s − 0.329i·11-s + (0.309 − 0.571i)12-s − 1.18·15-s + (0.546 + 0.837i)16-s − 1.09·17-s + (−0.141 + 0.560i)18-s + (0.871 − 1.61i)20-s + ⋯
L(s)  = 1  + (−0.245 + 0.969i)2-s + 0.649i·3-s + (−0.879 − 0.475i)4-s + 1.83i·5-s + (−0.629 − 0.159i)6-s + (0.677 − 0.735i)8-s + 0.578·9-s + (−1.77 − 0.449i)10-s − 0.329i·11-s + (0.309 − 0.571i)12-s − 1.18·15-s + (0.546 + 0.837i)16-s − 1.09·17-s + (−0.141 + 0.560i)18-s + (0.871 − 1.61i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2872\)    =    \(2^{3} \cdot 359\)
Sign: $-0.677 + 0.735i$
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: \((2,\ 2872,\ (\ :0),\ -0.677 + 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7436974711\)
\(L(\frac12)\) \(\approx\) \(0.7436974711\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.245 - 0.969i)T \)
359 \( 1 + T \)
good3 \( 1 - 0.649iT - T^{2} \)
5 \( 1 - 1.83iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 0.329iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 1.09T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.75T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.99iT - T^{2} \)
41 \( 1 + 0.165T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.57T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.490T + T^{2} \)
79 \( 1 - 0.803T + T^{2} \)
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

−9.657830559567978981768640663759, −8.520620175268250198750487388966, −7.84354726343949569457381649944, −7.01478633050042079900255379973, −6.52395229924627897022917614714, −5.91576055397878032130723478571, −4.77674262280104237722342551284, −3.99688305080165402270848422573, −3.25240561477521632175488386114, −1.96474740131150746514095683532, 0.50347692938117134674454567475, 1.69991618003679776808011601461, 2.15774378866898994505230756276, 3.90083997712648203798869044577, 4.35670539275448965592921699857, 5.13651897007615494511032846619, 6.06698640452842648696789121047, 7.26455048582189088212933768925, 8.016275861452499639440590404645, 8.526521275164567666928599919449

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