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Properties

Label 1-2888-2888.1141-r0-0-0
Degree $1$
Conductor $2888$
Sign $-0.998 - 0.0608i$
Analytic cond. $13.4118$
Root an. cond. $13.4118$
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  + (0.401 − 0.915i)3-s + (−0.245 − 0.969i)5-s + (0.789 − 0.614i)7-s + (−0.677 − 0.735i)9-s + (0.677 − 0.735i)11-s + (0.401 − 0.915i)13-s + (−0.986 − 0.164i)15-s + (0.789 − 0.614i)17-s + (−0.245 − 0.969i)21-s + (−0.401 − 0.915i)23-s + (−0.879 + 0.475i)25-s + (−0.945 + 0.324i)27-s + (−0.789 + 0.614i)29-s + (0.945 − 0.324i)31-s + (−0.401 − 0.915i)33-s + ⋯
L(s)  = 1  + (0.401 − 0.915i)3-s + (−0.245 − 0.969i)5-s + (0.789 − 0.614i)7-s + (−0.677 − 0.735i)9-s + (0.677 − 0.735i)11-s + (0.401 − 0.915i)13-s + (−0.986 − 0.164i)15-s + (0.789 − 0.614i)17-s + (−0.245 − 0.969i)21-s + (−0.401 − 0.915i)23-s + (−0.879 + 0.475i)25-s + (−0.945 + 0.324i)27-s + (−0.789 + 0.614i)29-s + (0.945 − 0.324i)31-s + (−0.401 − 0.915i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.998 - 0.0608i$
Analytic conductor: \(13.4118\)
Root analytic conductor: \(13.4118\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (1141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2888,\ (0:\ ),\ -0.998 - 0.0608i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06575727018 - 2.158234989i\)
\(L(\frac12)\) \(\approx\) \(0.06575727018 - 2.158234989i\)
\(L(1)\) \(\approx\) \(0.9587954028 - 0.9350105353i\)
\(L(1)\) \(\approx\) \(0.9587954028 - 0.9350105353i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.401 - 0.915i)T \)
5 \( 1 + (-0.245 - 0.969i)T \)
7 \( 1 + (0.789 - 0.614i)T \)
11 \( 1 + (0.677 - 0.735i)T \)
13 \( 1 + (0.401 - 0.915i)T \)
17 \( 1 + (0.789 - 0.614i)T \)
23 \( 1 + (-0.401 - 0.915i)T \)
29 \( 1 + (-0.789 + 0.614i)T \)
31 \( 1 + (0.945 - 0.324i)T \)
37 \( 1 + (0.677 - 0.735i)T \)
41 \( 1 + (-0.986 - 0.164i)T \)
43 \( 1 + (0.0825 + 0.996i)T \)
47 \( 1 + (-0.677 + 0.735i)T \)
53 \( 1 + (0.677 - 0.735i)T \)
59 \( 1 + (0.986 + 0.164i)T \)
61 \( 1 + (-0.546 + 0.837i)T \)
67 \( 1 + (-0.546 - 0.837i)T \)
71 \( 1 + (0.546 + 0.837i)T \)
73 \( 1 + (0.789 - 0.614i)T \)
79 \( 1 + (-0.0825 - 0.996i)T \)
83 \( 1 + (-0.245 + 0.969i)T \)
89 \( 1 + (0.789 + 0.614i)T \)
97 \( 1 + (0.546 - 0.837i)T \)
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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

−19.4493379109902301799917536051, −18.85546160005702985657572744926, −18.19235524469789382277148382092, −17.25103095132252470362617109390, −16.77971797208130766613939981946, −15.640981357690858962196761823722, −15.26165164637975451921125214771, −14.647789826160535124533504371599, −14.14173841750748855818491552576, −13.42868187898059030022062827440, −11.951222480952156349239523295221, −11.7115829868798941118073344131, −10.96973540797118658400125557083, −10.06894378964260648782026159562, −9.62652400305257956388818390992, −8.69019134950653110231256606329, −8.0640885391307270113139393209, −7.27710414459988726710072635706, −6.32507132282808731944304558782, −5.56056413279758969275343422862, −4.608493009682806852767406922195, −3.91834314536174622839062339637, −3.29400721753551249338819087115, −2.2176494360902219198544737624, −1.64270571363917132424607982959, 0.7231261141158092472083429288, 1.05414488349541322177818067632, 2.047302927711991736708534355063, 3.19694627513692265566266578062, 3.87560995358727725387405303886, 4.847426405130155405227850911600, 5.6649305042096330513997497040, 6.41409376105465221161862446459, 7.4582130484878459032489253122, 7.99479625540905626551244044271, 8.50565015923933066595901706544, 9.2220135680237208791281414492, 10.19594095281553166766119917230, 11.253597694589110568547554360981, 11.72796242507438540531372513409, 12.50907605761939823140420535233, 13.177067150550405659792143679881, 13.7890240415268367549426596745, 14.45615421791340804689004413880, 15.08688026384973900132881555919, 16.2784464832721941830938190562, 16.67206809363749358601230462335, 17.485766408517380280180981971777, 18.101676090970815422436637911478, 18.78568861766137270840140464963

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