Properties

Label 2-136-17.16-c3-0-3
Degree $2$
Conductor $136$
Sign $-0.366 - 0.930i$
Analytic cond. $8.02425$
Root an. cond. $2.83271$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.95i·3-s + 4.89i·5-s − 4.46i·7-s + 18.2·9-s + 60.3i·11-s − 56.1·13-s − 14.4·15-s + (25.6 + 65.2i)17-s − 134.·19-s + 13.1·21-s + 39.1i·23-s + 101.·25-s + 133. i·27-s − 113. i·29-s + 306. i·31-s + ⋯
L(s)  = 1  + 0.568i·3-s + 0.437i·5-s − 0.240i·7-s + 0.677·9-s + 1.65i·11-s − 1.19·13-s − 0.248·15-s + (0.366 + 0.930i)17-s − 1.62·19-s + 0.136·21-s + 0.355i·23-s + 0.808·25-s + 0.953i·27-s − 0.728i·29-s + 1.77i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.366 - 0.930i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.366 - 0.930i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.777347 + 1.14163i\)
\(L(\frac12)\) \(\approx\) \(0.777347 + 1.14163i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-25.6 - 65.2i)T \)
good3 \( 1 - 2.95iT - 27T^{2} \)
5 \( 1 - 4.89iT - 125T^{2} \)
7 \( 1 + 4.46iT - 343T^{2} \)
11 \( 1 - 60.3iT - 1.33e3T^{2} \)
13 \( 1 + 56.1T + 2.19e3T^{2} \)
19 \( 1 + 134.T + 6.85e3T^{2} \)
23 \( 1 - 39.1iT - 1.21e4T^{2} \)
29 \( 1 + 113. iT - 2.43e4T^{2} \)
31 \( 1 - 306. iT - 2.97e4T^{2} \)
37 \( 1 + 61.9iT - 5.06e4T^{2} \)
41 \( 1 + 317. iT - 6.89e4T^{2} \)
43 \( 1 - 122.T + 7.95e4T^{2} \)
47 \( 1 - 303.T + 1.03e5T^{2} \)
53 \( 1 - 133.T + 1.48e5T^{2} \)
59 \( 1 + 130.T + 2.05e5T^{2} \)
61 \( 1 + 772. iT - 2.26e5T^{2} \)
67 \( 1 - 378.T + 3.00e5T^{2} \)
71 \( 1 + 465. iT - 3.57e5T^{2} \)
73 \( 1 - 664. iT - 3.89e5T^{2} \)
79 \( 1 + 925. iT - 4.93e5T^{2} \)
83 \( 1 + 723.T + 5.71e5T^{2} \)
89 \( 1 - 889.T + 7.04e5T^{2} \)
97 \( 1 - 1.50e3iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.74576847995922363839665592578, −12.31499437803890662233088002899, −10.53340510222488740721881530569, −10.25702213384928793610702938187, −9.094718324648664240970943589486, −7.52717138842350186666638198775, −6.73244953962319434047911828522, −4.93637818565212394862529198136, −3.99223566076898249114010176156, −2.09525123158431134408737225818, 0.69861508580436482713111469346, 2.57644932046035899043491044522, 4.43344634261499444978101691324, 5.81372630496606405589608059469, 7.02289925298132202102218432095, 8.164609642179398448199750116739, 9.156891426823460969179645644050, 10.40073857943734867795624417472, 11.57417769729155294400124632091, 12.55871983967909768291998343143

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