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Properties

Label 1-2808-2808.1219-r1-0-0
Degree $1$
Conductor $2808$
Sign $-0.0453 + 0.998i$
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

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

Invariants

Degree: \(1\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $-0.0453 + 0.998i$
Analytic conductor: \(301.761\)
Root analytic conductor: \(301.761\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (1219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2808,\ (1:\ ),\ -0.0453 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7054021959 + 0.7381430595i\)
\(L(\frac12)\) \(\approx\) \(0.7054021959 + 0.7381430595i\)
\(L(1)\) \(\approx\) \(0.7579104210 + 0.09958009384i\)
\(L(1)\) \(\approx\) \(0.7579104210 + 0.09958009384i\)

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

−18.96770184487803275548631595756, −18.40086037502380207810734019295, −17.3545000754611350845981902932, −16.54223526647748166618941734670, −16.08390002263032556823057581530, −15.43880555631682873941599759904, −14.67766817852233409449440946722, −13.87272538723247219336659650325, −13.106848543256373959936294347322, −12.5609133707041631864731861670, −11.56129309609237965233821811341, −11.03768601589299278582340205771, −10.39516822548510644531237488353, −9.449630948541373401871856818139, −8.68679173239391859543575072207, −7.987063733362114007192584264677, −7.096722207128020898017892894769, −6.55200967814407126625573749658, −5.77826897375617896876794500456, −4.54114486318494299840423324286, −4.05541520062068135702240851616, −2.90609496401250412401713525936, −2.74351462918127337921152030878, −0.9433771332756383574070632546, −0.30331137253457304206828945030, 0.628848592285476929087819453900, 1.8343959090049233244136498194, 2.77345408839585959989200109216, 3.63248662071252156720173469835, 4.39940998089421223830254558435, 5.05647359404944280512743382488, 6.20208132093601389174789180644, 6.80456179592419101209070126635, 7.5791669507985656782826437317, 8.47408980190357712232204838356, 8.99670392638418311158429710876, 9.88935619232130236465949792302, 10.62810692574796944161860049560, 11.39298282191770595722347545224, 12.37777668302590531411537420693, 12.642870306627762833740912798115, 13.27787855048512176665290108338, 14.44677863288107886388872278771, 15.25219721852284404386625945901, 15.55400531136966817151408502406, 16.29544792728438878800514181476, 17.1284431242976641092060868795, 17.682465308870295898579399237093, 18.74542080739209811150424008823, 19.31685665807285361150362877333

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