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Properties

Label 2-168-168.101-c3-0-77
Degree $2$
Conductor $168$
Sign $0.676 + 0.736i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.17 − 1.80i)2-s + (4.78 + 2.03i)3-s + (1.48 − 7.86i)4-s + (11.2 − 6.48i)5-s + (14.0 − 4.20i)6-s + (−0.198 + 18.5i)7-s + (−10.9 − 19.7i)8-s + (18.7 + 19.4i)9-s + (12.7 − 34.3i)10-s + (11.4 − 19.7i)11-s + (23.0 − 34.5i)12-s − 3.71·13-s + (32.9 + 40.6i)14-s + (66.8 − 8.16i)15-s + (−59.5 − 23.3i)16-s + (−52.8 + 91.4i)17-s + ⋯
L(s)  = 1  + (0.769 − 0.638i)2-s + (0.920 + 0.391i)3-s + (0.185 − 0.982i)4-s + (1.00 − 0.579i)5-s + (0.958 − 0.285i)6-s + (−0.0107 + 0.999i)7-s + (−0.484 − 0.874i)8-s + (0.693 + 0.720i)9-s + (0.403 − 1.08i)10-s + (0.312 − 0.541i)11-s + (0.555 − 0.831i)12-s − 0.0792·13-s + (0.629 + 0.776i)14-s + (1.15 − 0.140i)15-s + (−0.931 − 0.364i)16-s + (−0.753 + 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.676 + 0.736i$
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: \((2,\ 168,\ (\ :3/2),\ 0.676 + 0.736i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.54815 - 1.55891i\)
\(L(\frac12)\) \(\approx\) \(3.54815 - 1.55891i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.17 + 1.80i)T \)
3 \( 1 + (-4.78 - 2.03i)T \)
7 \( 1 + (0.198 - 18.5i)T \)
good5 \( 1 + (-11.2 + 6.48i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-11.4 + 19.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 3.71T + 2.19e3T^{2} \)
17 \( 1 + (52.8 - 91.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (58.0 + 100. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-54.0 + 31.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 57.9T + 2.43e4T^{2} \)
31 \( 1 + (10.4 + 6.02i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-122. + 70.6i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 417.T + 6.89e4T^{2} \)
43 \( 1 + 212. iT - 7.95e4T^{2} \)
47 \( 1 + (-88.0 - 152. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (201. - 349. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (24.0 + 13.8i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-425. - 737. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (683. + 394. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 482. iT - 3.57e5T^{2} \)
73 \( 1 + (-228. - 131. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (71.1 + 123. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 477. iT - 5.71e5T^{2} \)
89 \( 1 + (-677. - 1.17e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.36e3iT - 9.12e5T^{2} \)
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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

−12.56145684861766789527945064382, −11.17867491447074112278992229928, −10.20065407564950082714940552539, −9.086455016073112139432214595879, −8.691620398381900323488709471127, −6.52268482243262899713381323124, −5.43008297472432535939882127991, −4.31979868177774894946555485892, −2.78141375058776095422154603983, −1.76196959828354058428095142471, 2.04590453576689331648919898853, 3.37749009590980761346609898660, 4.67797595079511736078657934503, 6.41140324825854634977431653095, 6.99272596354741865215514699630, 8.029534481883070777166905425620, 9.338150053037601197267884311589, 10.30580283391470790866937327534, 11.76082627137829871217416558276, 13.00371303827196880224968063173

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