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Properties

Label 1-5616-5616.1435-r0-0-0
Degree $1$
Conductor $5616$
Sign $-0.925 + 0.379i$
Analytic cond. $26.0805$
Root an. cond. $26.0805$
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 + (0.984 + 0.173i)7-s + (−0.939 + 0.342i)11-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.766 + 0.642i)25-s + (0.642 − 0.766i)29-s + (−0.984 + 0.173i)31-s + (−0.866 − 0.5i)35-s + (0.5 + 0.866i)37-s + (0.642 + 0.766i)41-s + (−0.342 − 0.939i)43-s + (−0.984 − 0.173i)47-s + (0.939 + 0.342i)49-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s + (0.984 + 0.173i)7-s + (−0.939 + 0.342i)11-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.766 + 0.642i)25-s + (0.642 − 0.766i)29-s + (−0.984 + 0.173i)31-s + (−0.866 − 0.5i)35-s + (0.5 + 0.866i)37-s + (0.642 + 0.766i)41-s + (−0.342 − 0.939i)43-s + (−0.984 − 0.173i)47-s + (0.939 + 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5616\)    =    \(2^{4} \cdot 3^{3} \cdot 13\)
Sign: $-0.925 + 0.379i$
Analytic conductor: \(26.0805\)
Root analytic conductor: \(26.0805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5616} (1435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5616,\ (0:\ ),\ -0.925 + 0.379i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08044159645 + 0.4077229781i\)
\(L(\frac12)\) \(\approx\) \(0.08044159645 + 0.4077229781i\)
\(L(1)\) \(\approx\) \(0.8224355639 + 0.06753940045i\)
\(L(1)\) \(\approx\) \(0.8224355639 + 0.06753940045i\)

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 \( 1 + (0.984 + 0.173i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.642 - 0.766i)T \)
31 \( 1 + (-0.984 + 0.173i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.984 - 0.173i)T \)
53 \( 1 + iT \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.984 - 0.173i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
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

−17.788077426062610059871001223112, −16.823720680253979094777616989923, −16.02194281607436441006378569009, −15.73633176233171529830464342815, −14.78275527264811080779549926145, −14.46522964254126672241497063673, −13.610393414363403040310692431183, −12.934893167807808186024094110312, −12.12364420581248730365286001058, −11.42803571577539875713054194804, −10.988789462147894822570027392867, −10.466505692635342214306785614439, −9.43488251796292209565232779651, −8.70336752065599324254642126669, −7.91215178705923908829047140432, −7.54913150548584215307060255459, −6.91980183032009399458748620742, −5.85475372088857481259324456620, −5.04147333330432134220503131210, −4.59554023934843619865862309596, −3.64169316630780272741556430270, −2.97540034862455988210902628750, −2.18085674008714954374266587284, −1.11345151845056922557811813505, −0.11858969858813619921805041028, 1.13965443358200597226395146868, 1.92446997709352867334134046490, 2.79576694251558703170000906171, 3.74687152357598686369550335366, 4.443033541835096782068294126794, 4.96028588974219106312623709534, 5.777658544072302271409937310519, 6.591623673320673142946500534782, 7.653708252128810435123133486224, 8.043251583094981624498204463981, 8.40040429256208601876204306267, 9.34966182775485347103503506257, 10.376854960626770142923367805367, 10.72359275121939777404277140911, 11.53624565399491435432855281191, 12.28120280291220295524560428709, 12.58950190462394093670507030610, 13.4449151204320137679784585416, 14.39231279903830159989219973329, 14.90872240499606998478004806454, 15.37224832119946102378025720606, 16.16983771198826994374775820561, 16.797442497503350901346525711628, 17.36953180153298997527463692423, 18.394754577913890699467111944

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