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Properties

Label 1-2888-2888.267-r1-0-0
Degree $1$
Conductor $2888$
Sign $-0.539 - 0.841i$
Analytic cond. $310.358$
Root an. cond. $310.358$
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)3-s + (0.401 − 0.915i)5-s + (−0.546 + 0.837i)7-s + (0.945 − 0.324i)9-s + (0.945 + 0.324i)11-s + (0.986 − 0.164i)13-s + (−0.245 + 0.969i)15-s + (0.546 − 0.837i)17-s + (0.401 − 0.915i)21-s + (0.986 + 0.164i)23-s + (−0.677 − 0.735i)25-s + (−0.879 + 0.475i)27-s + (−0.546 + 0.837i)29-s + (0.879 − 0.475i)31-s + (−0.986 − 0.164i)33-s + ⋯
L(s)  = 1  + (−0.986 + 0.164i)3-s + (0.401 − 0.915i)5-s + (−0.546 + 0.837i)7-s + (0.945 − 0.324i)9-s + (0.945 + 0.324i)11-s + (0.986 − 0.164i)13-s + (−0.245 + 0.969i)15-s + (0.546 − 0.837i)17-s + (0.401 − 0.915i)21-s + (0.986 + 0.164i)23-s + (−0.677 − 0.735i)25-s + (−0.879 + 0.475i)27-s + (−0.546 + 0.837i)29-s + (0.879 − 0.475i)31-s + (−0.986 − 0.164i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.539 - 0.841i$
Analytic conductor: \(310.358\)
Root analytic conductor: \(310.358\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2888,\ (1:\ ),\ -0.539 - 0.841i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6044681604 - 1.105440906i\)
\(L(\frac12)\) \(\approx\) \(0.6044681604 - 1.105440906i\)
\(L(1)\) \(\approx\) \(0.8755415142 - 0.1349857877i\)
\(L(1)\) \(\approx\) \(0.8755415142 - 0.1349857877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.986 + 0.164i)T \)
5 \( 1 + (0.401 - 0.915i)T \)
7 \( 1 + (-0.546 + 0.837i)T \)
11 \( 1 + (0.945 + 0.324i)T \)
13 \( 1 + (0.986 - 0.164i)T \)
17 \( 1 + (0.546 - 0.837i)T \)
23 \( 1 + (0.986 + 0.164i)T \)
29 \( 1 + (-0.546 + 0.837i)T \)
31 \( 1 + (0.879 - 0.475i)T \)
37 \( 1 + (-0.945 - 0.324i)T \)
41 \( 1 + (0.245 - 0.969i)T \)
43 \( 1 + (0.789 + 0.614i)T \)
47 \( 1 + (-0.945 - 0.324i)T \)
53 \( 1 + (-0.945 - 0.324i)T \)
59 \( 1 + (0.245 - 0.969i)T \)
61 \( 1 + (0.0825 - 0.996i)T \)
67 \( 1 + (-0.0825 - 0.996i)T \)
71 \( 1 + (0.0825 + 0.996i)T \)
73 \( 1 + (0.546 - 0.837i)T \)
79 \( 1 + (-0.789 - 0.614i)T \)
83 \( 1 + (-0.401 - 0.915i)T \)
89 \( 1 + (0.546 + 0.837i)T \)
97 \( 1 + (-0.0825 + 0.996i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.09580165051074514717828874483, −18.59075054498936835978407622910, −17.59355899939739939257897572580, −17.200610379371150281870026979065, −16.59036630877778595738241477765, −15.842306693679515319910676380438, −15.01140332829354525207127100505, −14.16211853225700452585371301751, −13.51361339282845521518864597003, −12.93826268052314796242245656152, −11.995191600976296725600732472722, −11.21925897286687536938892011286, −10.758343090757100982141792943964, −10.093463558176922337523439740, −9.4242234409184151604369752810, −8.31740458914113374980324903305, −7.31118462036558193534592381366, −6.6720670457339197837397499697, −6.22699737739225224613665538774, −5.58395989656249439324353155688, −4.32539578957472796847628854372, −3.73344611127168783368192709338, −2.89238281402977528432922703332, −1.48680311760784599431711150798, −1.03873097256555395513632900747, 0.27006557073503676272597895148, 1.13109796090874323037196688026, 1.85307576271086296384397596944, 3.19132452657819514958176163929, 4.009077471599115320384658002422, 5.041778431518509805336368584834, 5.40460830282256093810290501165, 6.29043960198077291890777664287, 6.76233992721259381074275781402, 7.90901435900414072023643225285, 9.02453107941391491575624292664, 9.29023332248875847708025884062, 10.03422696915661850606706730257, 11.04424405651448540314388042420, 11.690406012712590382173877237009, 12.39375037338373578972285676256, 12.8167189520675427601219223203, 13.618660491365610569715950759318, 14.56121146295896043505430495423, 15.54954922364872471676869837583, 16.024930289302899750209124068122, 16.58974906562422593308998160169, 17.35702018935605646040549384778, 17.82325610975208447383352272379, 18.73569071116260318438205573626

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