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Properties

Label 1-164-164.127-r1-0-0
Degree $1$
Conductor $164$
Sign $0.371 + 0.928i$
Analytic cond. $17.6242$
Root an. cond. $17.6242$
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  + 3-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)7-s + 9-s + (−0.809 + 0.587i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (0.309 + 0.951i)21-s + (−0.309 + 0.951i)23-s + (0.309 + 0.951i)25-s + 27-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + ⋯
L(s)  = 1  + 3-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)7-s + 9-s + (−0.809 + 0.587i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (0.309 + 0.951i)21-s + (−0.309 + 0.951i)23-s + (0.309 + 0.951i)25-s + 27-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(17.6242\)
Root analytic conductor: \(17.6242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 164,\ (1:\ ),\ 0.371 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.705949142 + 1.154385870i\)
\(L(\frac12)\) \(\approx\) \(1.705949142 + 1.154385870i\)
\(L(1)\) \(\approx\) \(1.305434374 + 0.2698946988i\)
\(L(1)\) \(\approx\) \(1.305434374 + 0.2698946988i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.309 + 0.951i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−26.84988193656552002772057157109, −26.57893076978957440862337484898, −25.55266918657572150240262501199, −24.25703025216693789425642248754, −23.60546836664371265286186269503, −22.47664694848959156999806982966, −21.22541022020181331787272764422, −20.316017950827103977939730066603, −19.510923089426949395009025692670, −18.687830497261086580785497196506, −17.533394361554235634374148609986, −16.056681149820301405094111029729, −15.26724213035640697965926389601, −14.28187809307685341152739261870, −13.44270668834955410300320317587, −12.231837552759635275170386696849, −10.70100712184384170773516417029, −10.156719918636994241476142345129, −8.328252518232304668310075909879, −7.84187734948546859512411906892, −6.76251761297895718625035973674, −4.81842060423521283153837677203, −3.574525371868600503598060777487, −2.70317370999533954943014734495, −0.69997780172213721729626841908, 1.59155812027550934524993373320, 2.90443604812176192847367333527, 4.25058406888734160487443073888, 5.34427190556672539317239348672, 7.29136090210127442024335564108, 8.07640305643811885258930313104, 9.04309245201448270155557782676, 10.02877872289077209491057976680, 11.823238497417977162564587654026, 12.39114051176523844767731751327, 13.70869254711813813640477690514, 14.78843465239249390345537579719, 15.61759437061663184659850938248, 16.42704987954222535142229390681, 18.12535795607838786279543824768, 18.93264705288176070896402499783, 19.79889066087708802059715799936, 20.89582813349031965183532848860, 21.38279019937452773302685461743, 22.96454460610099081146954424093, 23.97780880952731464203524883119, 24.7963031415056709897064868465, 25.638148576599009040147684402671, 26.70921086160640940588317164649, 27.59196929828343773906946099736

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