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Properties

Label 1-2808-2808.2045-r1-0-0
Degree $1$
Conductor $2808$
Sign $0.998 - 0.0453i$
Analytic cond. $301.761$
Root an. cond. $301.761$
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  + (0.939 − 0.342i)5-s + (0.939 + 0.342i)7-s + (−0.173 − 0.984i)11-s + (0.5 + 0.866i)17-s + 19-s + (0.939 − 0.342i)23-s + (0.766 − 0.642i)25-s + (−0.939 − 0.342i)29-s + (−0.173 + 0.984i)31-s + 35-s + 37-s + (0.173 − 0.984i)41-s + (−0.766 + 0.642i)43-s + (0.173 + 0.984i)47-s + (0.766 + 0.642i)49-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)5-s + (0.939 + 0.342i)7-s + (−0.173 − 0.984i)11-s + (0.5 + 0.866i)17-s + 19-s + (0.939 − 0.342i)23-s + (0.766 − 0.642i)25-s + (−0.939 − 0.342i)29-s + (−0.173 + 0.984i)31-s + 35-s + 37-s + (0.173 − 0.984i)41-s + (−0.766 + 0.642i)43-s + (0.173 + 0.984i)47-s + (0.766 + 0.642i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.998 - 0.0453i$
Analytic conductor: \(301.761\)
Root analytic conductor: \(301.761\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (2045, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2808,\ (1:\ ),\ 0.998 - 0.0453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.979135200 - 0.09025025194i\)
\(L(\frac12)\) \(\approx\) \(3.979135200 - 0.09025025194i\)
\(L(1)\) \(\approx\) \(1.587085153 - 0.06053660946i\)
\(L(1)\) \(\approx\) \(1.587085153 - 0.06053660946i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−18.657892414590698569956036233293, −18.34549568054840808962337553263, −17.686334244312591416213603753329, −17.01493737448913710648956175369, −16.43754274677757943088832731783, −15.25983088909599032404855083594, −14.77602716502318749647268354460, −14.136852459874929542787691794641, −13.39048912271997673264171875426, −12.85067471607522728014883601693, −11.66579106976673641658474837067, −11.30937766684569390226903806095, −10.31487674777251741512869080311, −9.71833918891440872958278963078, −9.164334003932329969832148814876, −8.03564460942209810520186904778, −7.28965610410690816726812045774, −6.85936950135587588978352606845, −5.52122457829325157076050075336, −5.25029338803152904730930971212, −4.32543449757164381752050922765, −3.25432216983985386375892338309, −2.360021594385220424615290495617, −1.62425192436677646780337035980, −0.77088754116893007645297352040, 0.859915530972862710517751382113, 1.4427383539107876485502839421, 2.41114606884829730166131060705, 3.23127116727746628317346049343, 4.29910785914779079023989575571, 5.33700217857965507146658335002, 5.55851021731390486120733731391, 6.455712081864927108293084100092, 7.514395474129402967016641747598, 8.27869273944360844603182405156, 8.90518859470170265304788323813, 9.57904315240344708657969255032, 10.54990269421199523120132506393, 11.06202651248590924471746335118, 11.90929461186148999356702485500, 12.70887634806782256266547529845, 13.41098286959834077627770350248, 14.092598684474327109657118382, 14.672022841873277466435688652034, 15.42883301272453778562070233839, 16.50869144043894195062508425394, 16.799893965323076111023972432388, 17.72958842922667792783375720536, 18.21012485621086200466884522695, 18.90132848086136139472205354557

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