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Properties

Label 1-2808-2808.2165-r0-0-0
Degree $1$
Conductor $2808$
Sign $-0.425 - 0.904i$
Analytic cond. $13.0402$
Root an. cond. $13.0402$
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  + (0.642 + 0.766i)5-s + (−0.984 − 0.173i)7-s + (−0.984 − 0.173i)11-s + 17-s + (−0.866 + 0.5i)19-s + (0.173 + 0.984i)23-s + (−0.173 + 0.984i)25-s + (−0.939 + 0.342i)29-s + (0.342 − 0.939i)31-s + (−0.5 − 0.866i)35-s + (−0.866 − 0.5i)37-s + (−0.642 − 0.766i)41-s + (−0.939 + 0.342i)43-s + (−0.342 − 0.939i)47-s + (0.939 + 0.342i)49-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)5-s + (−0.984 − 0.173i)7-s + (−0.984 − 0.173i)11-s + 17-s + (−0.866 + 0.5i)19-s + (0.173 + 0.984i)23-s + (−0.173 + 0.984i)25-s + (−0.939 + 0.342i)29-s + (0.342 − 0.939i)31-s + (−0.5 − 0.866i)35-s + (−0.866 − 0.5i)37-s + (−0.642 − 0.766i)41-s + (−0.939 + 0.342i)43-s + (−0.342 − 0.939i)47-s + (0.939 + 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $-0.425 - 0.904i$
Analytic conductor: \(13.0402\)
Root analytic conductor: \(13.0402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (2165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2808,\ (0:\ ),\ -0.425 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2135055053 - 0.3363594969i\)
\(L(\frac12)\) \(\approx\) \(0.2135055053 - 0.3363594969i\)
\(L(1)\) \(\approx\) \(0.8321288125 + 0.06913576357i\)
\(L(1)\) \(\approx\) \(0.8321288125 + 0.06913576357i\)

Euler product

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

−19.37001787538374852850996233955, −18.73566865732640840180180862507, −18.0838049959908307390383265014, −17.157714835621093933131957461203, −16.65554073560067557448513236509, −16.02005304675241788564556976776, −15.30511746322367783984786249171, −14.49313306674761670722696378000, −13.52338489821349244674760679178, −13.03717833403343437659212956023, −12.52058466538446354105081371151, −11.80037350007914335605292956225, −10.53651048177609249862680388536, −10.14953704503093671472808519074, −9.40540978392171236390039114029, −8.62920663194863593987223843690, −8.020901428784060481054400251089, −6.88792232724422121080620235339, −6.29393076257806535617922663577, −5.37089895060889868548292959076, −4.907453751556217597110097073604, −3.81497559236390731817419558, −2.857543809328145587670789815466, −2.17681799849972985338321877508, −1.06046191743275358364830514006, 0.12551537136996667145070891259, 1.638361846315906653109768685909, 2.46255695205091720754418646897, 3.328404335945093232100755181386, 3.81835365489202229205685497608, 5.32265566350668718406833486175, 5.69378022791204576545085621605, 6.58433955742529320864003434003, 7.25643857964601053768238114599, 7.99632375419602825743366159984, 9.02602046834042125304096741980, 9.846796412945608991052247580007, 10.28266090196352133186753878144, 10.90797236891580949345373712905, 11.89859228823414056387437359889, 12.72452327880471266990949984652, 13.45734112790266537217626124674, 13.78560887970564954869279370280, 15.020258850597421708286642838046, 15.16801393448581346573981230417, 16.38817596318105865820829950575, 16.72765224839146801583511360955, 17.64637173744046007669458026847, 18.3798970750196287916628259569, 18.996738268668385298719172258786

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