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Properties

Label 1-416-416.155-r1-0-0
Degree $1$
Conductor $416$
Sign $-0.195 + 0.980i$
Analytic cond. $44.7054$
Root an. cond. $44.7054$
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  + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s − i·7-s − i·9-s + (−0.707 − 0.707i)11-s − 15-s − 17-s + (0.707 − 0.707i)19-s + (0.707 + 0.707i)21-s + i·23-s + i·25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s − 33-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s − i·7-s − i·9-s + (−0.707 − 0.707i)11-s − 15-s − 17-s + (0.707 − 0.707i)19-s + (0.707 + 0.707i)21-s + i·23-s + i·25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.195 + 0.980i$
Analytic conductor: \(44.7054\)
Root analytic conductor: \(44.7054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 416,\ (1:\ ),\ -0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2500598928 + 0.3046988609i\)
\(L(\frac12)\) \(\approx\) \(0.2500598928 + 0.3046988609i\)
\(L(1)\) \(\approx\) \(0.8956250948 - 0.2279387425i\)
\(L(1)\) \(\approx\) \(0.8956250948 - 0.2279387425i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 - T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 - iT \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 - T \)
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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

−23.74301669072559504259172431499, −22.58662912318505361390736650444, −22.43795774984357803541728597203, −20.80148224814367863324709446895, −20.52288259368019042663081623861, −19.5657820091723986048678607074, −18.79017828354039917382162432670, −17.71550646352646054790278597557, −16.60727652984886590331215890112, −15.68378540865385366572806797831, −15.12531798330966317657946920026, −14.13358844979155680184802415911, −13.46923472231912568697620989232, −12.191316452218439935748446848492, −10.90007454032673920514120148452, −10.434431610928953545413241850541, −9.518386914255768016023207771545, −8.16810343506782610728441735166, −7.58277550402416986994773174703, −6.58928633613555875369525110130, −4.87338401389884239261740604613, −4.10970228929099986076988613347, −3.20607303789261306976570047275, −2.10146281785943143109448288586, −0.096658260223438690849549260669, 1.25769852304453404668606689330, 2.560508380344564610204990520052, 3.454175156650168457109989166151, 4.85753990886725580448838958946, 5.88277974071372162259511883738, 7.14913010534958570119623046147, 8.0802930662951984660859223865, 8.77161696498500156410223779830, 9.47659692889825909722294732904, 11.23526440361952495757735659801, 11.86779094478806781952924825566, 12.95140457691569219669974503689, 13.38510867819372038736135229226, 14.65975243806533150637372796718, 15.60600481526477130256937426853, 16.055240533726592270833162382295, 17.58687965131872886342021282307, 18.28087718272715552783508803282, 19.27892843244658448721879148099, 19.72028240401701731499337004688, 20.76679966208325644130590316159, 21.48802793319155880458505453473, 22.65325212501853979687581762530, 23.778316332839052006465144020975, 24.33520413220710748628422265175

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