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Properties

Label 2-168-56.3-c3-0-1
Degree $2$
Conductor $168$
Sign $-0.0691 - 0.997i$
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  + (−2.82 − 0.128i)2-s + (−2.59 − 1.5i)3-s + (7.96 + 0.724i)4-s + (−8.82 − 15.2i)5-s + (7.14 + 4.57i)6-s + (−18.4 + 1.45i)7-s + (−22.4 − 3.07i)8-s + (4.5 + 7.79i)9-s + (22.9 + 44.3i)10-s + (24.2 − 41.9i)11-s + (−19.6 − 13.8i)12-s − 47.7·13-s + (52.3 − 1.73i)14-s + 52.9i·15-s + (62.9 + 11.5i)16-s + (1.25 + 0.724i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0453i)2-s + (−0.499 − 0.288i)3-s + (0.995 + 0.0906i)4-s + (−0.788 − 1.36i)5-s + (0.486 + 0.311i)6-s + (−0.996 + 0.0784i)7-s + (−0.990 − 0.135i)8-s + (0.166 + 0.288i)9-s + (0.726 + 1.40i)10-s + (0.663 − 1.14i)11-s + (−0.471 − 0.332i)12-s − 1.01·13-s + (0.999 − 0.0331i)14-s + 0.911i·15-s + (0.983 + 0.180i)16-s + (0.0178 + 0.0103i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0691 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0691 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.0691 - 0.997i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ -0.0691 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0300284 + 0.0321803i\)
\(L(\frac12)\) \(\approx\) \(0.0300284 + 0.0321803i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.82 + 0.128i)T \)
3 \( 1 + (2.59 + 1.5i)T \)
7 \( 1 + (18.4 - 1.45i)T \)
good5 \( 1 + (8.82 + 15.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-24.2 + 41.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 47.7T + 2.19e3T^{2} \)
17 \( 1 + (-1.25 - 0.724i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-19.1 + 11.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-58.0 + 33.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 218. iT - 2.43e4T^{2} \)
31 \( 1 + (151. - 263. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-61.8 + 35.7i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 419. iT - 6.89e4T^{2} \)
43 \( 1 - 258.T + 7.95e4T^{2} \)
47 \( 1 + (246. + 426. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (18.8 + 10.8i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (371. + 214. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-206. - 357. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (108. - 187. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 813. iT - 3.57e5T^{2} \)
73 \( 1 + (715. + 413. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-614. + 354. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 464. iT - 5.71e5T^{2} \)
89 \( 1 + (191. - 110. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.56e3iT - 9.12e5T^{2} \)
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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

−12.33748878806668280815203602630, −11.69551785594368131628344867826, −10.59035010178111609596411684882, −9.259984596968118150554838113651, −8.716511626585789759332708309597, −7.50554381074365941084094035773, −6.46166605612490232180174092680, −5.12964786490348171279196410680, −3.31999656760637153441541330836, −1.08716851969097536735995144281, 0.03527327564040784393758035074, 2.56717940431033938710201211387, 3.94859578196426706414774521734, 6.05872927575101640128382099492, 7.07931507441323672207733863467, 7.53973228094227823904973137272, 9.464784732028454965004526026095, 9.918897697160374077782418269578, 10.96883896644950542705844932068, 11.76091248145790034819434815687

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