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Properties

Label 1-2888-2888.1331-r1-0-0
Degree $1$
Conductor $2888$
Sign $0.708 - 0.705i$
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  + (−0.677 + 0.735i)3-s + (0.879 + 0.475i)5-s + (−0.245 − 0.969i)7-s + (−0.0825 − 0.996i)9-s + (−0.0825 + 0.996i)11-s + (0.677 − 0.735i)13-s + (−0.945 + 0.324i)15-s + (0.245 + 0.969i)17-s + (0.879 + 0.475i)21-s + (0.677 + 0.735i)23-s + (0.546 + 0.837i)25-s + (0.789 + 0.614i)27-s + (−0.245 − 0.969i)29-s + (−0.789 − 0.614i)31-s + (−0.677 − 0.735i)33-s + ⋯
L(s)  = 1  + (−0.677 + 0.735i)3-s + (0.879 + 0.475i)5-s + (−0.245 − 0.969i)7-s + (−0.0825 − 0.996i)9-s + (−0.0825 + 0.996i)11-s + (0.677 − 0.735i)13-s + (−0.945 + 0.324i)15-s + (0.245 + 0.969i)17-s + (0.879 + 0.475i)21-s + (0.677 + 0.735i)23-s + (0.546 + 0.837i)25-s + (0.789 + 0.614i)27-s + (−0.245 − 0.969i)29-s + (−0.789 − 0.614i)31-s + (−0.677 − 0.735i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.708 - 0.705i$
Analytic conductor: \(310.358\)
Root analytic conductor: \(310.358\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (1331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2888,\ (1:\ ),\ 0.708 - 0.705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.547171228 - 0.6388884048i\)
\(L(\frac12)\) \(\approx\) \(1.547171228 - 0.6388884048i\)
\(L(1)\) \(\approx\) \(0.9868049735 + 0.1312972098i\)
\(L(1)\) \(\approx\) \(0.9868049735 + 0.1312972098i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.677 + 0.735i)T \)
5 \( 1 + (0.879 + 0.475i)T \)
7 \( 1 + (-0.245 - 0.969i)T \)
11 \( 1 + (-0.0825 + 0.996i)T \)
13 \( 1 + (0.677 - 0.735i)T \)
17 \( 1 + (0.245 + 0.969i)T \)
23 \( 1 + (0.677 + 0.735i)T \)
29 \( 1 + (-0.245 - 0.969i)T \)
31 \( 1 + (-0.789 - 0.614i)T \)
37 \( 1 + (0.0825 - 0.996i)T \)
41 \( 1 + (0.945 - 0.324i)T \)
43 \( 1 + (-0.986 - 0.164i)T \)
47 \( 1 + (0.0825 - 0.996i)T \)
53 \( 1 + (0.0825 - 0.996i)T \)
59 \( 1 + (0.945 - 0.324i)T \)
61 \( 1 + (0.401 - 0.915i)T \)
67 \( 1 + (-0.401 - 0.915i)T \)
71 \( 1 + (0.401 + 0.915i)T \)
73 \( 1 + (0.245 + 0.969i)T \)
79 \( 1 + (0.986 + 0.164i)T \)
83 \( 1 + (-0.879 + 0.475i)T \)
89 \( 1 + (0.245 - 0.969i)T \)
97 \( 1 + (-0.401 + 0.915i)T \)
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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

−18.87756874741778635322601835609, −18.167837423159997362164687971157, −18.02906282874898655426230211620, −16.66565691457877106623552249881, −16.55311736768960159303067550734, −15.8903492103336112981253295962, −14.65060477483406645985226881677, −13.93103642014679237938162280270, −13.324798202688364704512494058942, −12.71277767799735051254665114984, −12.03239258973471514699542191697, −11.301409509122981012563748331503, −10.689256359309924901158454939292, −9.64535270985798629474619428734, −8.835197814491572832937001348070, −8.4987155546175587832404015969, −7.286097405778961759704583857335, −6.46219903210191921014047922971, −5.96432136194182252415061772441, −5.29134204802153306453690213238, −4.661612534462050229055979071051, −3.1604656685191205169588396915, −2.477306818122787467861088330460, −1.49308300957592936048600217711, −0.86879239163516952983221529223, 0.3437501499540889446506038968, 1.33431436276792072738811623645, 2.32100231883372899746689287824, 3.63560453747391317639416636269, 3.84606491920302746111029713581, 5.078881003285746330819826052249, 5.63883063099351817837846790245, 6.42951388349047416935680109411, 7.078004557811075137032982427751, 7.949057139836018264560585105535, 9.14071913637704916563542959428, 9.82704352817926120804934415070, 10.258168847576521525243399740849, 10.89924436839880777769386787664, 11.48789590930475997985302931131, 12.82436093146875143608420406126, 12.98768182734848987769281433331, 14.01105609592597330272961204596, 14.8468187628194739440081163048, 15.27564740862536927040564723464, 16.16079681581193128853130198558, 16.99988166309434329160961821227, 17.38604467798148026801296214653, 17.90785139790741456835433824148, 18.71483800491778206208539965431

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