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Properties

Label 2-136-136.3-c3-0-42
Degree $2$
Conductor $136$
Sign $0.872 + 0.488i$
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  + (2.61 − 1.08i)2-s + (6.61 + 1.31i)3-s + (5.65 − 5.65i)4-s + (18.7 − 3.72i)6-s + (8.65 − 20.9i)8-s + (17.0 + 7.06i)9-s + (0.752 + 3.78i)11-s + (44.8 − 29.9i)12-s − 64i·16-s + (−21.0 + 66.8i)17-s + 52.2·18-s + (−13.9 + 5.75i)19-s + (6.06 + 9.07i)22-s + (84.7 − 126. i)24-s + (−47.8 + 115. i)25-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (1.27 + 0.253i)3-s + (0.707 − 0.707i)4-s + (1.27 − 0.253i)6-s + (0.382 − 0.923i)8-s + (0.632 + 0.261i)9-s + (0.0206 + 0.103i)11-s + (1.07 − 0.720i)12-s − i·16-s + (−0.299 + 0.954i)17-s + 0.684·18-s + (−0.167 + 0.0695i)19-s + (0.0587 + 0.0879i)22-s + (0.720 − 1.07i)24-s + (−0.382 + 0.923i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.84726 - 1.00468i\)
\(L(\frac12)\) \(\approx\) \(3.84726 - 1.00468i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.61 + 1.08i)T \)
17 \( 1 + (21.0 - 66.8i)T \)
good3 \( 1 + (-6.61 - 1.31i)T + (24.9 + 10.3i)T^{2} \)
5 \( 1 + (47.8 - 115. i)T^{2} \)
7 \( 1 + (131. + 316. i)T^{2} \)
11 \( 1 + (-0.752 - 3.78i)T + (-1.22e3 + 509. i)T^{2} \)
13 \( 1 - 2.19e3iT^{2} \)
19 \( 1 + (13.9 - 5.75i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-1.12e4 + 4.65e3i)T^{2} \)
29 \( 1 + (-9.33e3 + 2.25e4i)T^{2} \)
31 \( 1 + (2.75e4 + 1.14e4i)T^{2} \)
37 \( 1 + (-4.67e4 - 1.93e4i)T^{2} \)
41 \( 1 + (-87.4 + 130. i)T + (-2.63e4 - 6.36e4i)T^{2} \)
43 \( 1 + (76.5 + 31.6i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 1.03e5iT^{2} \)
53 \( 1 + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (251. - 607. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-8.68e4 - 2.09e5i)T^{2} \)
67 \( 1 - 984. iT - 3.00e5T^{2} \)
71 \( 1 + (-3.30e5 - 1.36e5i)T^{2} \)
73 \( 1 + (107. + 160. i)T + (-1.48e5 + 3.59e5i)T^{2} \)
79 \( 1 + (4.55e5 - 1.88e5i)T^{2} \)
83 \( 1 + (-43.2 - 104. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-1.02e3 + 1.02e3i)T - 7.04e5iT^{2} \)
97 \( 1 + (-917. + 613. i)T + (3.49e5 - 8.43e5i)T^{2} \)
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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

−12.97761874461765008672981257455, −11.79576158864393030911930278923, −10.63146526149395037353414976869, −9.623123875449216101528631590313, −8.521738376897006033842340491728, −7.24960754596058112305453499203, −5.81680972391487880242567662172, −4.26967496049585492246611299754, −3.26209276480564124413282098225, −1.93776099907990458255380651761, 2.26145158752685293539806171221, 3.37037106913313037504067966842, 4.74617849143301133931754180455, 6.30724708987728427402673094523, 7.50493805649044576651428719183, 8.320912139624336420672109937538, 9.439769865004670356495802054233, 11.01774366802318935363045873266, 12.14571080381903911213283656054, 13.17776457660938457881415018658

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