Properties

Label 2-136-17.16-c3-0-0
Degree $2$
Conductor $136$
Sign $0.372 - 0.928i$
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  − 9.26i·3-s + 16.4i·5-s + 34.5i·7-s − 58.7·9-s + 7.42i·11-s − 42.0·13-s + 151.·15-s + (−26.0 + 65.0i)17-s + 59.9·19-s + 319.·21-s + 49.4i·23-s − 144.·25-s + 294. i·27-s − 259. i·29-s − 92.2i·31-s + ⋯
L(s)  = 1  − 1.78i·3-s + 1.46i·5-s + 1.86i·7-s − 2.17·9-s + 0.203i·11-s − 0.896·13-s + 2.61·15-s + (−0.372 + 0.928i)17-s + 0.723·19-s + 3.32·21-s + 0.448i·23-s − 1.15·25-s + 2.09i·27-s − 1.66i·29-s − 0.534i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.372 - 0.928i$
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: \((2,\ 136,\ (\ :3/2),\ 0.372 - 0.928i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.915480 + 0.619244i\)
\(L(\frac12)\) \(\approx\) \(0.915480 + 0.619244i\)
\(L(\frac{5}{2})\) 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 \( 1 + (26.0 - 65.0i)T \)
good3 \( 1 + 9.26iT - 27T^{2} \)
5 \( 1 - 16.4iT - 125T^{2} \)
7 \( 1 - 34.5iT - 343T^{2} \)
11 \( 1 - 7.42iT - 1.33e3T^{2} \)
13 \( 1 + 42.0T + 2.19e3T^{2} \)
19 \( 1 - 59.9T + 6.85e3T^{2} \)
23 \( 1 - 49.4iT - 1.21e4T^{2} \)
29 \( 1 + 259. iT - 2.43e4T^{2} \)
31 \( 1 + 92.2iT - 2.97e4T^{2} \)
37 \( 1 - 207. iT - 5.06e4T^{2} \)
41 \( 1 - 176. iT - 6.89e4T^{2} \)
43 \( 1 + 19.0T + 7.95e4T^{2} \)
47 \( 1 - 80.1T + 1.03e5T^{2} \)
53 \( 1 - 319.T + 1.48e5T^{2} \)
59 \( 1 - 11.0T + 2.05e5T^{2} \)
61 \( 1 - 712. iT - 2.26e5T^{2} \)
67 \( 1 - 484.T + 3.00e5T^{2} \)
71 \( 1 + 443. iT - 3.57e5T^{2} \)
73 \( 1 - 337. iT - 3.89e5T^{2} \)
79 \( 1 - 840. iT - 4.93e5T^{2} \)
83 \( 1 - 456.T + 5.71e5T^{2} \)
89 \( 1 + 1.20e3T + 7.04e5T^{2} \)
97 \( 1 + 638. iT - 9.12e5T^{2} \)
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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

−12.82938778358412499500677169745, −11.89561172547254941015719182888, −11.43720668445035139650945934400, −9.780091063381266607234291876389, −8.383990901346061204934039108047, −7.45461853698736173845145159682, −6.44669743628292844935642238731, −5.71400502052104329540434254131, −2.80061386230783044682825253551, −2.11174470696619767331386013847, 0.53926807483611096241002500927, 3.56600680122728331184887710562, 4.63310322515273990066849221236, 5.15093853223361305431554412119, 7.29036317821476901918417651024, 8.705614401392361533952211589774, 9.530148666255956401861039080227, 10.34339968845934339060030177574, 11.22362191270355462942603810655, 12.53315395242571226980320122948

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