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Properties

Label 1-2808-2808.173-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  + (−0.766 − 0.642i)5-s + (−0.766 + 0.642i)7-s + (0.939 + 0.342i)11-s + (0.5 + 0.866i)17-s + 19-s + (−0.766 − 0.642i)23-s + (0.173 + 0.984i)25-s + (0.766 − 0.642i)29-s + (0.939 − 0.342i)31-s + 35-s + 37-s + (−0.939 + 0.342i)41-s + (−0.173 − 0.984i)43-s + (−0.939 − 0.342i)47-s + (0.173 − 0.984i)49-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)5-s + (−0.766 + 0.642i)7-s + (0.939 + 0.342i)11-s + (0.5 + 0.866i)17-s + 19-s + (−0.766 − 0.642i)23-s + (0.173 + 0.984i)25-s + (0.766 − 0.642i)29-s + (0.939 − 0.342i)31-s + 35-s + 37-s + (−0.939 + 0.342i)41-s + (−0.173 − 0.984i)43-s + (−0.939 − 0.342i)47-s + (0.173 − 0.984i)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: $\chi_{2808} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2808,\ (1:\ ),\ 0.998 - 0.0453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.776857625 - 0.04030067848i\)
\(L(\frac12)\) \(\approx\) \(1.776857625 - 0.04030067848i\)
\(L(1)\) \(\approx\) \(0.9565135207 + 0.01090813224i\)
\(L(1)\) \(\approx\) \(0.9565135207 + 0.01090813224i\)

Euler product

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

−19.21463692251374627190258694057, −18.313377624182967978951696650948, −17.79556626886498353317866009083, −16.68711237988946388976940426597, −16.243582208685303623692288288965, −15.660429988399537505649874220851, −14.741748039501147412372322303226, −14.00210914123781619390334281869, −13.633470054271555571235760032812, −12.50850614087963669701555205188, −11.72776147206341648814328877923, −11.42518419001817613046647949766, −10.29704760023216247719251178729, −9.84983871555410970110499015652, −9.00224414485630351893380844621, −7.99791964856221597772835897341, −7.36384884878502732578750133278, −6.684382239541094661474416026074, −6.070936218830247071885921028853, −4.90261516929867816644954152738, −4.040809302522163459526367175804, −3.29465047119556511371380517574, −2.86941222870869369921067130632, −1.32907771248686694667276610214, −0.55681505136530790153320729502, 0.53897999211486312781906546534, 1.41310550173560298538186560169, 2.52399888927457942174555092420, 3.499547723201723327052972246271, 4.10135655863000859667957734807, 4.95552407339033464418741703696, 5.90755026964227699553477296108, 6.52494122439578423771230706465, 7.45061747724002602805565911039, 8.32051926024687519361155841426, 8.78954323075426200791132743841, 9.764103862776794966728213234881, 10.155774710702830752775074728073, 11.56777993136520921764258635945, 11.89994640119446795463653788243, 12.47584938025968903051857094559, 13.24628759030031558209903968157, 14.070005030411151404763895796758, 15.07549468710778768110679536515, 15.38670137848503193226274926612, 16.4213163997187796475771376507, 16.61235807421726403413975661859, 17.59787852340917560394972401982, 18.42545483268258791640102902646, 19.18319391202439110461676469161

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