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Properties

Label 1-236-236.139-r1-0-0
Degree $1$
Conductor $236$
Sign $0.986 + 0.163i$
Analytic cond. $25.3617$
Root an. cond. $25.3617$
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  + (0.725 + 0.687i)3-s + (0.976 + 0.214i)5-s + (−0.796 − 0.605i)7-s + (0.0541 + 0.998i)9-s + (0.370 − 0.928i)11-s + (0.0541 − 0.998i)13-s + (0.561 + 0.827i)15-s + (0.796 − 0.605i)17-s + (0.947 + 0.319i)19-s + (−0.161 − 0.986i)21-s + (0.856 + 0.515i)23-s + (0.907 + 0.419i)25-s + (−0.647 + 0.762i)27-s + (0.468 − 0.883i)29-s + (0.947 − 0.319i)31-s + ⋯
L(s)  = 1  + (0.725 + 0.687i)3-s + (0.976 + 0.214i)5-s + (−0.796 − 0.605i)7-s + (0.0541 + 0.998i)9-s + (0.370 − 0.928i)11-s + (0.0541 − 0.998i)13-s + (0.561 + 0.827i)15-s + (0.796 − 0.605i)17-s + (0.947 + 0.319i)19-s + (−0.161 − 0.986i)21-s + (0.856 + 0.515i)23-s + (0.907 + 0.419i)25-s + (−0.647 + 0.762i)27-s + (0.468 − 0.883i)29-s + (0.947 − 0.319i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 236 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(236\)    =    \(2^{2} \cdot 59\)
Sign: $0.986 + 0.163i$
Analytic conductor: \(25.3617\)
Root analytic conductor: \(25.3617\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{236} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 236,\ (1:\ ),\ 0.986 + 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.937515518 + 0.2420815877i\)
\(L(\frac12)\) \(\approx\) \(2.937515518 + 0.2420815877i\)
\(L(1)\) \(\approx\) \(1.626922602 + 0.1731891301i\)
\(L(1)\) \(\approx\) \(1.626922602 + 0.1731891301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
59 \( 1 \)
good3 \( 1 + (0.725 + 0.687i)T \)
5 \( 1 + (0.976 + 0.214i)T \)
7 \( 1 + (-0.796 - 0.605i)T \)
11 \( 1 + (0.370 - 0.928i)T \)
13 \( 1 + (0.0541 - 0.998i)T \)
17 \( 1 + (0.796 - 0.605i)T \)
19 \( 1 + (0.947 + 0.319i)T \)
23 \( 1 + (0.856 + 0.515i)T \)
29 \( 1 + (0.468 - 0.883i)T \)
31 \( 1 + (0.947 - 0.319i)T \)
37 \( 1 + (-0.994 + 0.108i)T \)
41 \( 1 + (-0.856 + 0.515i)T \)
43 \( 1 + (0.370 + 0.928i)T \)
47 \( 1 + (-0.976 + 0.214i)T \)
53 \( 1 + (0.267 - 0.963i)T \)
61 \( 1 + (0.468 + 0.883i)T \)
67 \( 1 + (0.994 + 0.108i)T \)
71 \( 1 + (-0.976 + 0.214i)T \)
73 \( 1 + (-0.161 + 0.986i)T \)
79 \( 1 + (0.725 - 0.687i)T \)
83 \( 1 + (-0.647 - 0.762i)T \)
89 \( 1 + (0.468 - 0.883i)T \)
97 \( 1 + (-0.161 - 0.986i)T \)
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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

−25.78406825322062440122589178374, −25.18025015885983787436156634619, −24.44415181764542594494428058391, −23.324925381449122885504164813696, −22.23232004321918754540646765594, −21.257377467489673705654975786894, −20.418834163915944209741772453897, −19.361387878284781104130870589488, −18.63115723746885650290740443413, −17.68496241215392807717532002855, −16.71743181379658488321371268535, −15.44684190081652455595535281549, −14.387512690841182065750423289811, −13.65998633025413248063001903165, −12.58822969501525484364092745695, −12.0566842695931106194739773247, −10.154183636910934137319261154615, −9.312481127875424450195112019601, −8.64883199308616779081059594026, −7.04120636126207704717535540938, −6.42994308477746504519577020129, −5.09358437289833219551581137440, −3.4204447438487175282142937824, −2.286729420690678075157755011428, −1.27278713578677873460100424016, 1.04794791435228716707872355697, 2.91006455982391350143695407176, 3.41285472661391119187570585551, 5.06303231696657528307286996243, 6.08834757124484256452328285491, 7.42242778383565825865559889417, 8.61395158743172516782610424616, 9.84668903254946013247419400236, 10.0769202958337833270702356130, 11.40305899784763906192861177469, 13.110845491446466025691123404969, 13.7081153295000306774537036003, 14.49045994443465915365008463418, 15.71972175731669874087702081143, 16.536264044877979480472525800241, 17.44416468821016000089095806941, 18.7791749351287369193060390882, 19.574690851099358332587585959700, 20.64685372574879590424633283532, 21.25517142921304528474446143664, 22.37045392719387379247331693917, 22.89111477786406410577803466348, 24.664011739416466922835219226755, 25.16565359974892683849656315592, 26.09967012123994005083770242617

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