Properties

Label 1-136-136.123-r1-0-0
Degree $1$
Conductor $136$
Sign $-0.615 + 0.788i$
Analytic cond. $14.6152$
Root an. cond. $14.6152$
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

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

Invariants

Degree: \(1\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(14.6152\)
Root analytic conductor: \(14.6152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 136,\ (1:\ ),\ -0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1244085814 + 0.2549730460i\)
\(L(\frac12)\) \(\approx\) \(0.1244085814 + 0.2549730460i\)
\(L(1)\) \(\approx\) \(0.7562392846 - 0.09309731024i\)
\(L(1)\) \(\approx\) \(0.7562392846 - 0.09309731024i\)

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

−27.82504338723185267207222564188, −27.193822419109076730640981265444, −25.98512353975013284860802099677, −24.91740120256258420739250296296, −24.11389004717864291094513537351, −22.76630963246001423230377628276, −21.47271057968734680001685849438, −21.35073432623792008854955937332, −19.92241414184827828293630217699, −19.072490053002757122939496225051, −17.42607737772377709945380944377, −16.644407438116307485888610784281, −15.66989486813702223080374997402, −14.84024408056652347995717538485, −13.469803705937317740161242268056, −12.18280093889166845585878679334, −11.28740536958546263643245160091, −9.812818860013439778864366977729, −8.97953508914503090503183097847, −8.10043650763573390395062648611, −5.92096161389584698528753326987, −5.14210601404982872516384870184, −3.90238058266905516886037157021, −2.35445857209522629260005694542, −0.10372809833489118836417972618, 1.784015543364541407517245955370, 3.04805211923289649127034345846, 4.71833646969700179773710723961, 6.6053588888178704636890472471, 7.05258467856932466483888982366, 8.16744448133686566713741015148, 9.94182714268389593654001028000, 10.8683501482153879505123690854, 12.14303732091477948746194391775, 13.116938921949845919190250858904, 14.32599569840794610875084201913, 14.89373687984412042205469295243, 16.75208098295581665954050539984, 17.60543688332156693772491301777, 18.474010895365753997386107376749, 19.57040846179680500423416212598, 20.23660712108612491568848843037, 21.83575209075905887093265807413, 22.97650114860515592637497043705, 23.4342613348185567419911073280, 24.672720736643936846803644111256, 25.70790317199763028338367592667, 26.44398363855975214857106611054, 27.569647433710978931284273505463, 28.97782180986109345333196202712

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