↑

Properties

Label 2-168-168.101-c3-0-21
Degree $2$
Conductor $168$
Sign $-0.375 + 0.926i$
Analytic cond. $9.91232$
Root an. cond. $3.14838$
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  + (0.133 + 2.82i)2-s + (2.59 + 4.50i)3-s + (−7.96 + 0.754i)4-s + (−12.7 + 7.37i)5-s + (−12.3 + 7.92i)6-s + (−2.36 + 18.3i)7-s + (−3.19 − 22.4i)8-s + (−13.5 + 23.3i)9-s + (−22.5 − 35.0i)10-s + (19.8 − 34.3i)11-s + (−24.0 − 33.9i)12-s + 55.6·13-s + (−52.2 − 4.23i)14-s + (−66.2 − 38.4i)15-s + (62.8 − 12.0i)16-s + (−2.03 + 3.52i)17-s + ⋯
L(s)  = 1  + (0.0472 + 0.998i)2-s + (0.498 + 0.866i)3-s + (−0.995 + 0.0943i)4-s + (−1.14 + 0.659i)5-s + (−0.842 + 0.538i)6-s + (−0.127 + 0.991i)7-s + (−0.141 − 0.989i)8-s + (−0.502 + 0.864i)9-s + (−0.712 − 1.10i)10-s + (0.542 − 0.940i)11-s + (−0.578 − 0.815i)12-s + 1.18·13-s + (−0.996 − 0.0808i)14-s + (−1.14 − 0.661i)15-s + (0.982 − 0.187i)16-s + (−0.0290 + 0.0503i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.375 + 0.926i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ -0.375 + 0.926i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.511442 - 0.759051i\)
\(L(\frac12)\) \(\approx\) \(0.511442 - 0.759051i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.133 - 2.82i)T \)
3 \( 1 + (-2.59 - 4.50i)T \)
7 \( 1 + (2.36 - 18.3i)T \)
good5 \( 1 + (12.7 - 7.37i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-19.8 + 34.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 55.6T + 2.19e3T^{2} \)
17 \( 1 + (2.03 - 3.52i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (42.3 + 73.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (182. - 105. i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 50.0T + 2.43e4T^{2} \)
31 \( 1 + (-209. - 120. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (108. - 62.9i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 221.T + 6.89e4T^{2} \)
43 \( 1 - 66.2iT - 7.95e4T^{2} \)
47 \( 1 + (27.6 + 47.8i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-69.8 + 121. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (411. + 237. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-184. - 319. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-669. - 386. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.00e3iT - 3.57e5T^{2} \)
73 \( 1 + (-632. - 365. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (7.31 + 12.6i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 367. iT - 5.71e5T^{2} \)
89 \( 1 + (224. + 388. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 602. iT - 9.12e5T^{2} \)
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

−13.44198220749523730026760269337, −11.87233706520800054178388505591, −11.04770734304740553255622826493, −9.714009470988397575450784269169, −8.519905920043019535623254663411, −8.247484873122646349883918156852, −6.66955550421969276513569691968, −5.57711333775821078966156241677, −4.06279713768384442729555190370, −3.26566339338321278013183151505, 0.40920664399826306237829867115, 1.71739009078218311042441457135, 3.69614582392399687463718813539, 4.29131473859066383319769599993, 6.39397855576656190005199373338, 7.890714999483698047369687383472, 8.382212741934992278474686026820, 9.650590520749376121558249685124, 10.81839604856857518340231525950, 12.05400732391255062305590987903

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