↑

Properties

Label 2-136-8.5-c3-0-40
Degree $2$
Conductor $136$
Sign $-0.533 + 0.845i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.67 − 0.932i)2-s − 3.64i·3-s + (6.25 − 4.98i)4-s − 5.60i·5-s + (−3.40 − 9.73i)6-s − 30.5·7-s + (12.0 − 19.1i)8-s + 13.6·9-s + (−5.22 − 14.9i)10-s − 28.1i·11-s + (−18.1 − 22.8i)12-s − 4.49i·13-s + (−81.6 + 28.5i)14-s − 20.4·15-s + (14.3 − 62.3i)16-s + 17·17-s + ⋯
L(s)  = 1  + (0.944 − 0.329i)2-s − 0.701i·3-s + (0.782 − 0.622i)4-s − 0.500i·5-s + (−0.231 − 0.662i)6-s − 1.65·7-s + (0.533 − 0.845i)8-s + 0.507·9-s + (−0.165 − 0.472i)10-s − 0.771i·11-s + (−0.437 − 0.549i)12-s − 0.0958i·13-s + (−1.55 + 0.544i)14-s − 0.351·15-s + (0.224 − 0.974i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.533 + 0.845i$
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: \((2,\ 136,\ (\ :3/2),\ -0.533 + 0.845i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.20066 - 2.17637i\)
\(L(\frac12)\) \(\approx\) \(1.20066 - 2.17637i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.67 + 0.932i)T \)
17 \( 1 - 17T \)
good3 \( 1 + 3.64iT - 27T^{2} \)
5 \( 1 + 5.60iT - 125T^{2} \)
7 \( 1 + 30.5T + 343T^{2} \)
11 \( 1 + 28.1iT - 1.33e3T^{2} \)
13 \( 1 + 4.49iT - 2.19e3T^{2} \)
19 \( 1 - 107. iT - 6.85e3T^{2} \)
23 \( 1 - 4.25T + 1.21e4T^{2} \)
29 \( 1 + 254. iT - 2.43e4T^{2} \)
31 \( 1 - 142.T + 2.97e4T^{2} \)
37 \( 1 - 176. iT - 5.06e4T^{2} \)
41 \( 1 - 290.T + 6.89e4T^{2} \)
43 \( 1 - 334. iT - 7.95e4T^{2} \)
47 \( 1 - 13.2T + 1.03e5T^{2} \)
53 \( 1 - 152. iT - 1.48e5T^{2} \)
59 \( 1 - 372. iT - 2.05e5T^{2} \)
61 \( 1 - 783. iT - 2.26e5T^{2} \)
67 \( 1 + 1.03e3iT - 3.00e5T^{2} \)
71 \( 1 - 137.T + 3.57e5T^{2} \)
73 \( 1 - 783.T + 3.89e5T^{2} \)
79 \( 1 + 971.T + 4.93e5T^{2} \)
83 \( 1 - 36.6iT - 5.71e5T^{2} \)
89 \( 1 - 276.T + 7.04e5T^{2} \)
97 \( 1 + 1.32e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.60061896918554020932725326699, −11.87658604603839082213600242213, −10.37578257985284140422826653094, −9.558373782835872801735289122127, −7.86836195584488162989523178142, −6.54278963733695330960195146690, −5.89405680353545697350014313702, −4.14767557526285358841834367471, −2.85520084921171653762346352688, −0.974789838697712492902663207101, 2.80200375472410361640486203812, 3.87494116026410296698135755200, 5.12032400911183231593651186835, 6.65738688378972425663532203847, 7.13926818265252819504130504570, 9.100001015377125730988621771778, 10.11616457908997114182246965846, 11.01109190216257162639306433579, 12.51647459438139654371277061402, 12.93716296540622064476138280891

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