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Properties

Label 2-136-136.3-c1-0-2
Degree $2$
Conductor $136$
Sign $0.539 - 0.842i$
Analytic cond. $1.08596$
Root an. cond. $1.04209$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 0.541i)2-s + (−3.17 − 0.630i)3-s + (1.41 − 1.41i)4-s + (4.48 − 0.892i)6-s + (−1.08 + 2.61i)8-s + (6.89 + 2.85i)9-s + (1.04 + 5.23i)11-s + (−5.37 + 3.59i)12-s − 4i·16-s + (3.85 + 1.46i)17-s − 10.5·18-s + (4.23 − 1.75i)19-s + (−4.19 − 6.27i)22-s + (5.08 − 7.60i)24-s + (−1.91 + 4.61i)25-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)2-s + (−1.83 − 0.364i)3-s + (0.707 − 0.707i)4-s + (1.83 − 0.364i)6-s + (−0.382 + 0.923i)8-s + (2.29 + 0.951i)9-s + (0.313 + 1.57i)11-s + (−1.55 + 1.03i)12-s − i·16-s + (0.934 + 0.355i)17-s − 2.48·18-s + (0.971 − 0.402i)19-s + (−0.893 − 1.33i)22-s + (1.03 − 1.55i)24-s + (−0.382 + 0.923i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.539 - 0.842i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.539 - 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.356906 + 0.195225i\)
\(L(\frac12)\) \(\approx\) \(0.356906 + 0.195225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.541i)T \)
17 \( 1 + (-3.85 - 1.46i)T \)
good3 \( 1 + (3.17 + 0.630i)T + (2.77 + 1.14i)T^{2} \)
5 \( 1 + (1.91 - 4.61i)T^{2} \)
7 \( 1 + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (-1.04 - 5.23i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 - 13iT^{2} \)
19 \( 1 + (-4.23 + 1.75i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (6.81 - 10.1i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (-11.5 - 4.77i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.0502 + 0.121i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-23.3 - 56.3i)T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (0.749 + 1.12i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (1.63 + 3.94i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-13.2 + 13.2i)T - 89iT^{2} \)
97 \( 1 + (-7.11 + 4.75i)T + (37.1 - 89.6i)T^{2} \)
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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

−13.00125830400459965594182379179, −11.98240264225714604821382145012, −11.39306492331975200921575924052, −10.22872840423006770973723648106, −9.590563620350448911267729367142, −7.63807084522341418593813919636, −6.96779803427106827762576484492, −5.86526603978362069651694538888, −4.86018931396894971854764098565, −1.42791268339009044775333028186, 0.803130704621839629332318522429, 3.67703664544189989768355551706, 5.50184266676704327507019112591, 6.38935824487168307668690561610, 7.68601048267618288488796434574, 9.197692147998894437201337318434, 10.26083142523162196591260908132, 10.95777140152628124195574671881, 11.83586694591341009254691610950, 12.34166672623494878654389198992

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