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Properties

Label 2-2872-2872.717-c0-0-7
Degree $2$
Conductor $2872$
Sign $0.789 - 0.614i$
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  + (0.677 + 0.735i)2-s − 1.67i·3-s + (−0.0825 + 0.996i)4-s + 0.649i·5-s + (1.23 − 1.13i)6-s + (−0.789 + 0.614i)8-s − 1.80·9-s + (−0.477 + 0.439i)10-s + 0.951i·11-s + (1.66 + 0.138i)12-s + 1.08·15-s + (−0.986 − 0.164i)16-s + 1.97·17-s + (−1.22 − 1.32i)18-s + (−0.647 − 0.0536i)20-s + ⋯
L(s)  = 1  + (0.677 + 0.735i)2-s − 1.67i·3-s + (−0.0825 + 0.996i)4-s + 0.649i·5-s + (1.23 − 1.13i)6-s + (−0.789 + 0.614i)8-s − 1.80·9-s + (−0.477 + 0.439i)10-s + 0.951i·11-s + (1.66 + 0.138i)12-s + 1.08·15-s + (−0.986 − 0.164i)16-s + 1.97·17-s + (−1.22 − 1.32i)18-s + (−0.647 − 0.0536i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2872\)    =    \(2^{3} \cdot 359\)
Sign: $0.789 - 0.614i$
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: \((2,\ 2872,\ (\ :0),\ 0.789 - 0.614i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.713902842\)
\(L(\frac12)\) \(\approx\) \(1.713902842\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.677 - 0.735i)T \)
359 \( 1 + T \)
good3 \( 1 + 1.67iT - T^{2} \)
5 \( 1 - 0.649iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 0.951iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 1.97T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 0.165T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.93iT - T^{2} \)
41 \( 1 - 0.490T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 0.803T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.35T + T^{2} \)
79 \( 1 + 1.89T + T^{2} \)
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

−8.533350935653728466240596119894, −7.950022945326213625886929534031, −7.17055268050982269781997890838, −7.06023320868138615127856136924, −6.07250925886780739825856667482, −5.55853392241203946752895949765, −4.46866048620890461193040252827, −3.22073343544648402547199734988, −2.61001464650444071727125508692, −1.40429590200286120261381336293, 0.990336949709210884526503798794, 2.66454227847824489173324545226, 3.50523985971312190798079117499, 4.02287309319329309190212704521, 4.88076230974178166710135644591, 5.60725473297114211350579499170, 5.88768464170596842624168617039, 7.44925065563096218094932784274, 8.607047464804094074376649952697, 9.073757503443671994871671751837

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