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Properties

Label 1-136-136.21-r0-0-0
Degree $1$
Conductor $136$
Sign $-0.615 + 0.788i$
Analytic cond. $0.631581$
Root an. cond. $0.631581$
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  + i·3-s + i·5-s + i·7-s − 9-s − i·11-s − 13-s − 15-s + 19-s − 21-s + i·23-s − 25-s − i·27-s + i·29-s − i·31-s + 33-s + ⋯
L(s)  = 1  + i·3-s + i·5-s + i·7-s − 9-s − i·11-s − 13-s − 15-s + 19-s − 21-s + i·23-s − 25-s − i·27-s + i·29-s − i·31-s + 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(0.631581\)
Root analytic conductor: \(0.631581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 136,\ (0:\ ),\ -0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4087145829 + 0.8376528448i\)
\(L(\frac12)\) \(\approx\) \(0.4087145829 + 0.8376528448i\)
\(L(1)\) \(\approx\) \(0.7806431473 + 0.5560712396i\)
\(L(1)\) \(\approx\) \(0.7806431473 + 0.5560712396i\)

Euler product

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

−28.51779131752332692563464911779, −27.18654131700025104059146920957, −26.08797106177436584960980718673, −24.89173223573886864111811392596, −24.32054919538588393351576622266, −23.33251620963818006262145973726, −22.50246412837844030951679885845, −20.755590257158385731205420180065, −20.08815476272668414850780826039, −19.298189148741528713528262597768, −17.77151341897512047935544456955, −17.22743926731785910956250862036, −16.1625488042386817060677610163, −14.55448885793690941835979261095, −13.58805059061768402974590721165, −12.58270772576064027850688191748, −11.91214094814135023150340351195, −10.2926646398968117696752557674, −9.057107062330786134481625458475, −7.70882305045389090108934442042, −7.040793847977333169796204133725, −5.42970964535123239345737497762, −4.22878634740939491678471843206, −2.28354825650837503999912319327, −0.85438846170893333754868343115, 2.601140858761200920153019914370, 3.47466835740831037531991200803, 5.1605932443441033981593373412, 6.09240108846985399816500219729, 7.70009400316001882514280162411, 9.07822801770358325864410053857, 9.96820472068977070634244581562, 11.16616327705763980134360292861, 11.90558689320776492047747729872, 13.74134755877533758738554507815, 14.74105859847167602091932242103, 15.46445673502547967105320068713, 16.49369773455989836520049340487, 17.75918661752074164509371954308, 18.823579735772636228378275135554, 19.77923755206464117633833211575, 21.175587759330768656821887016506, 22.06976343881646645922264463743, 22.3473915824378515684777908381, 23.874819425738983983105409012692, 25.114410927761518327924956082861, 26.08562924073127730831256197983, 26.951609311557284678427505185982, 27.62333022667567454284419631033, 28.88101738392273559308423789449

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