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Properties

Label 2-136-8.5-c3-0-26
Degree $2$
Conductor $136$
Sign $0.265 + 0.964i$
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  + (−2.56 − 1.18i)2-s + 7.88i·3-s + (5.17 + 6.10i)4-s − 14.0i·5-s + (9.38 − 20.2i)6-s − 13.6·7-s + (−6.01 − 21.8i)8-s − 35.2·9-s + (−16.6 + 36.0i)10-s + 23.7i·11-s + (−48.1 + 40.7i)12-s − 81.1i·13-s + (35.0 + 16.2i)14-s + 110.·15-s + (−10.5 + 63.1i)16-s + 17·17-s + ⋯
L(s)  = 1  + (−0.907 − 0.420i)2-s + 1.51i·3-s + (0.646 + 0.762i)4-s − 1.25i·5-s + (0.638 − 1.37i)6-s − 0.736·7-s + (−0.265 − 0.964i)8-s − 1.30·9-s + (−0.527 + 1.13i)10-s + 0.651i·11-s + (−1.15 + 0.981i)12-s − 1.73i·13-s + (0.668 + 0.309i)14-s + 1.90·15-s + (−0.164 + 0.986i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.265 + 0.964i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 0.265 + 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.550471 - 0.419286i\)
\(L(\frac12)\) \(\approx\) \(0.550471 - 0.419286i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.56 + 1.18i)T \)
17 \( 1 - 17T \)
good3 \( 1 - 7.88iT - 27T^{2} \)
5 \( 1 + 14.0iT - 125T^{2} \)
7 \( 1 + 13.6T + 343T^{2} \)
11 \( 1 - 23.7iT - 1.33e3T^{2} \)
13 \( 1 + 81.1iT - 2.19e3T^{2} \)
19 \( 1 + 92.7iT - 6.85e3T^{2} \)
23 \( 1 - 106.T + 1.21e4T^{2} \)
29 \( 1 + 174. iT - 2.43e4T^{2} \)
31 \( 1 + 144.T + 2.97e4T^{2} \)
37 \( 1 + 317. iT - 5.06e4T^{2} \)
41 \( 1 - 405.T + 6.89e4T^{2} \)
43 \( 1 + 180. iT - 7.95e4T^{2} \)
47 \( 1 + 114.T + 1.03e5T^{2} \)
53 \( 1 - 373. iT - 1.48e5T^{2} \)
59 \( 1 + 176. iT - 2.05e5T^{2} \)
61 \( 1 - 562. iT - 2.26e5T^{2} \)
67 \( 1 + 933. iT - 3.00e5T^{2} \)
71 \( 1 + 830.T + 3.57e5T^{2} \)
73 \( 1 + 234.T + 3.89e5T^{2} \)
79 \( 1 + 317.T + 4.93e5T^{2} \)
83 \( 1 + 198. iT - 5.71e5T^{2} \)
89 \( 1 + 1.54e3T + 7.04e5T^{2} \)
97 \( 1 - 180.T + 9.12e5T^{2} \)
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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

−12.54589667024312366577444553851, −11.10737798511247901161175245727, −10.25512783745520424155042703477, −9.396278104742046871750724752894, −8.877142798369000370779563985070, −7.54331207346843542916569354112, −5.57952085355068422278873770618, −4.33984355812074511674705092319, −2.99810983620824649486930992467, −0.47764099000829924776946176359, 1.52252065603131283494683382607, 2.96684897361984131602676863759, 6.02621300307777791292835822275, 6.74818272014348455736782746519, 7.29273639427226649262063238006, 8.536133679483398193572609292769, 9.725849515626157985109004071016, 10.98866534797449193322628970334, 11.73379595884881125585415824879, 12.97752313433857325193988065133

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