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Properties

Label 1-116-116.27-r0-0-0
Degree $1$
Conductor $116$
Sign $0.982 + 0.184i$
Analytic cond. $0.538701$
Root an. cond. $0.538701$
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  + (0.433 + 0.900i)3-s + (0.222 − 0.974i)5-s + (0.900 − 0.433i)7-s + (−0.623 + 0.781i)9-s + (0.781 − 0.623i)11-s + (−0.623 − 0.781i)13-s + (0.974 − 0.222i)15-s + i·17-s + (−0.433 + 0.900i)19-s + (0.781 + 0.623i)21-s + (0.222 + 0.974i)23-s + (−0.900 − 0.433i)25-s + (−0.974 − 0.222i)27-s + (0.974 + 0.222i)31-s + (0.900 + 0.433i)33-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)3-s + (0.222 − 0.974i)5-s + (0.900 − 0.433i)7-s + (−0.623 + 0.781i)9-s + (0.781 − 0.623i)11-s + (−0.623 − 0.781i)13-s + (0.974 − 0.222i)15-s + i·17-s + (−0.433 + 0.900i)19-s + (0.781 + 0.623i)21-s + (0.222 + 0.974i)23-s + (−0.900 − 0.433i)25-s + (−0.974 − 0.222i)27-s + (0.974 + 0.222i)31-s + (0.900 + 0.433i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.982 + 0.184i$
Analytic conductor: \(0.538701\)
Root analytic conductor: \(0.538701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 116,\ (0:\ ),\ 0.982 + 0.184i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.291281368 + 0.1204190117i\)
\(L(\frac12)\) \(\approx\) \(1.291281368 + 0.1204190117i\)
\(L(1)\) \(\approx\) \(1.256040464 + 0.1080161512i\)
\(L(1)\) \(\approx\) \(1.256040464 + 0.1080161512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.433 + 0.900i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
11 \( 1 + (0.781 - 0.623i)T \)
13 \( 1 + (-0.623 - 0.781i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.433 + 0.900i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (0.974 + 0.222i)T \)
37 \( 1 + (-0.781 - 0.623i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.974 + 0.222i)T \)
47 \( 1 + (-0.781 + 0.623i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.433 - 0.900i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.974 + 0.222i)T \)
79 \( 1 + (-0.781 - 0.623i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (-0.974 - 0.222i)T \)
97 \( 1 + (-0.433 + 0.900i)T \)
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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

−29.48360227956585231994636340270, −28.30940406475475086967376377003, −27.009637383663362881258935774599, −26.10534408381551946269726966084, −25.02846821564674238009839803045, −24.37186829577882329628175213789, −23.14686896313988452152292835744, −22.131174246143491002799175635771, −20.975436512267224163927717854436, −19.76321167189778366999991906682, −18.781961173757773076599924059049, −17.98524318496162305018587624926, −17.11062846223796557298682386063, −15.10122219366361704313680043637, −14.49839891404709582886531736055, −13.59255926357890745608906613774, −12.050223110318964133934174218835, −11.37863786092859696458466416187, −9.69866686225926759340034670904, −8.52650571305329622142934965295, −7.168968470640475931138209378537, −6.51125189799273858891965036889, −4.70187233851303144866635319443, −2.80808783475273120358663924669, −1.8286269253483285339095507707, 1.63040382159040981660255100511, 3.6016652982376083166478654972, 4.686261059949848908478068908325, 5.75729174613644229138860376899, 7.917315598875789225451069180554, 8.66319469671253649868305000509, 9.87276973946002017042500014616, 10.93136651081751586444536335294, 12.261784071760612529580817218850, 13.66979881204084066905820622511, 14.565045992816850932774426319224, 15.64642253848334282904862192788, 16.97005362698027519881399763330, 17.326704308558287153355522208588, 19.34622174090990916711473222486, 20.14902851581031978027205954978, 21.11799174667900264628391067669, 21.7477137935826762516394847555, 23.15691986309107960973006048336, 24.45686913772932337993264853915, 25.08492778503443075879232432448, 26.39781712849093075985826479871, 27.534448389608043670595119246160, 27.75928719178915121285744110233, 29.2562692246267287832949603725

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