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Properties

Label 2-136-136.67-c2-0-8
Degree $2$
Conductor $136$
Sign $-0.585 - 0.810i$
Analytic cond. $3.70573$
Root an. cond. $1.92502$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 + 1.17i)2-s + 2.59i·3-s + (1.21 + 3.80i)4-s − 3.44·5-s + (−3.05 + 4.18i)6-s − 1.33·7-s + (−2.52 + 7.59i)8-s + 2.27·9-s + (−5.55 − 4.05i)10-s + 7.47i·11-s + (−9.87 + 3.15i)12-s − 2.46i·13-s + (−2.14 − 1.56i)14-s − 8.92i·15-s + (−13.0 + 9.28i)16-s + (9.94 − 13.7i)17-s + ⋯
L(s)  = 1  + (0.807 + 0.589i)2-s + 0.864i·3-s + (0.304 + 0.952i)4-s − 0.688·5-s + (−0.509 + 0.698i)6-s − 0.190·7-s + (−0.315 + 0.948i)8-s + 0.253·9-s + (−0.555 − 0.405i)10-s + 0.679i·11-s + (−0.823 + 0.263i)12-s − 0.189i·13-s + (−0.153 − 0.112i)14-s − 0.594i·15-s + (−0.814 + 0.580i)16-s + (0.584 − 0.811i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.585 - 0.810i$
Analytic conductor: \(3.70573\)
Root analytic conductor: \(1.92502\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1),\ -0.585 - 0.810i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.883446 + 1.72687i\)
\(L(\frac12)\) \(\approx\) \(0.883446 + 1.72687i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.61 - 1.17i)T \)
17 \( 1 + (-9.94 + 13.7i)T \)
good3 \( 1 - 2.59iT - 9T^{2} \)
5 \( 1 + 3.44T + 25T^{2} \)
7 \( 1 + 1.33T + 49T^{2} \)
11 \( 1 - 7.47iT - 121T^{2} \)
13 \( 1 + 2.46iT - 169T^{2} \)
19 \( 1 - 4.36T + 361T^{2} \)
23 \( 1 - 33.2T + 529T^{2} \)
29 \( 1 - 16.2T + 841T^{2} \)
31 \( 1 - 34.6T + 961T^{2} \)
37 \( 1 + 2.64T + 1.36e3T^{2} \)
41 \( 1 + 29.6iT - 1.68e3T^{2} \)
43 \( 1 - 25.5T + 1.84e3T^{2} \)
47 \( 1 + 53.6iT - 2.20e3T^{2} \)
53 \( 1 - 0.647iT - 2.80e3T^{2} \)
59 \( 1 - 25.8T + 3.48e3T^{2} \)
61 \( 1 + 55.9T + 3.72e3T^{2} \)
67 \( 1 - 4.84T + 4.48e3T^{2} \)
71 \( 1 + 7.70T + 5.04e3T^{2} \)
73 \( 1 + 96.8iT - 5.32e3T^{2} \)
79 \( 1 + 100.T + 6.24e3T^{2} \)
83 \( 1 + 58.3T + 6.88e3T^{2} \)
89 \( 1 - 39.3T + 7.92e3T^{2} \)
97 \( 1 + 22.9iT - 9.40e3T^{2} \)
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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

−13.40213567847422800319205920604, −12.37183096144106242600147918549, −11.53099291082130552040342287422, −10.28710364927445789005953369556, −9.123059254613181935972546024515, −7.75805118637678584864079556418, −6.86356973459145163314542662382, −5.22060561730174825151096031031, −4.33368455309026012686914605576, −3.15293136319774133693941212854, 1.16252842375900373561959906323, 3.04059190213763567407518423123, 4.39946617258086375829067597309, 5.97016801311869840996511861348, 7.00443040004599054188783338632, 8.198854629648300179736680290822, 9.746765255770389769998165023687, 10.94107597344342000966580499215, 11.84020550015324091460133267274, 12.64620937841972767112916689232

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