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Properties

Label 2-2872-2872.717-c0-0-22
Degree $2$
Conductor $2872$
Sign $-0.879 + 0.475i$
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  + (0.986 − 0.164i)2-s − 1.47i·3-s + (0.945 − 0.324i)4-s − 1.93i·5-s + (−0.242 − 1.45i)6-s + (0.879 − 0.475i)8-s − 1.16·9-s + (−0.319 − 1.91i)10-s + 1.83i·11-s + (−0.477 − 1.39i)12-s − 2.85·15-s + (0.789 − 0.614i)16-s − 1.57·17-s + (−1.14 + 0.191i)18-s + (−0.629 − 1.83i)20-s + ⋯
L(s)  = 1  + (0.986 − 0.164i)2-s − 1.47i·3-s + (0.945 − 0.324i)4-s − 1.93i·5-s + (−0.242 − 1.45i)6-s + (0.879 − 0.475i)8-s − 1.16·9-s + (−0.319 − 1.91i)10-s + 1.83i·11-s + (−0.477 − 1.39i)12-s − 2.85·15-s + (0.789 − 0.614i)16-s − 1.57·17-s + (−1.14 + 0.191i)18-s + (−0.629 − 1.83i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2872\)    =    \(2^{3} \cdot 359\)
Sign: $-0.879 + 0.475i$
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: \((2,\ 2872,\ (\ :0),\ -0.879 + 0.475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.295323516\)
\(L(\frac12)\) \(\approx\) \(2.295323516\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.986 + 0.164i)T \)
359 \( 1 + T \)
good3 \( 1 + 1.47iT - T^{2} \)
5 \( 1 + 1.93iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.83iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 1.57T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.89T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.67iT - T^{2} \)
41 \( 1 - 1.09T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 0.165T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.97T + T^{2} \)
79 \( 1 + 0.490T + T^{2} \)
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

−8.565294000093989284000431441689, −7.60075823696682591614596413492, −7.09694433929947738725427298986, −6.39506862234131008152179070343, −5.42372666476090981716564991264, −4.64579465316316284641327656216, −4.32442148821095582807536918947, −2.60789322330350629548875023118, −1.79453973012996529124245118818, −1.11472902265782623534808621231, 2.51202945544431361041272885557, 3.04883445092255631581338479550, 3.72079545924387940196429963588, 4.37388094861512626140651406505, 5.50973475527180753641272387648, 6.03248930604133473354350588863, 6.83692299827700069724282261098, 7.45829261785413706880906069273, 8.653859955697990558451219312592, 9.275221887631953069081919237238

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