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Properties

Label 2-168-56.27-c3-0-13
Degree $2$
Conductor $168$
Sign $0.992 + 0.120i$
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  + (−0.713 − 2.73i)2-s − 3i·3-s + (−6.98 + 3.90i)4-s − 14.5·5-s + (−8.21 + 2.13i)6-s + (11.7 + 14.3i)7-s + (15.6 + 16.3i)8-s − 9·9-s + (10.3 + 39.8i)10-s − 28.6·11-s + (11.7 + 20.9i)12-s + 83.2·13-s + (30.8 − 42.3i)14-s + 43.6i·15-s + (33.5 − 54.5i)16-s + 27.2i·17-s + ⋯
L(s)  = 1  + (−0.252 − 0.967i)2-s − 0.577i·3-s + (−0.872 + 0.487i)4-s − 1.30·5-s + (−0.558 + 0.145i)6-s + (0.633 + 0.773i)7-s + (0.692 + 0.721i)8-s − 0.333·9-s + (0.328 + 1.26i)10-s − 0.784·11-s + (0.281 + 0.503i)12-s + 1.77·13-s + (0.589 − 0.807i)14-s + 0.752i·15-s + (0.523 − 0.851i)16-s + 0.388i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ 0.992 + 0.120i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.920164 - 0.0555442i\)
\(L(\frac12)\) \(\approx\) \(0.920164 - 0.0555442i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.713 + 2.73i)T \)
3 \( 1 + 3iT \)
7 \( 1 + (-11.7 - 14.3i)T \)
good5 \( 1 + 14.5T + 125T^{2} \)
11 \( 1 + 28.6T + 1.33e3T^{2} \)
13 \( 1 - 83.2T + 2.19e3T^{2} \)
17 \( 1 - 27.2iT - 4.91e3T^{2} \)
19 \( 1 + 62.8iT - 6.85e3T^{2} \)
23 \( 1 - 200. iT - 1.21e4T^{2} \)
29 \( 1 + 32.4iT - 2.43e4T^{2} \)
31 \( 1 - 51.2T + 2.97e4T^{2} \)
37 \( 1 - 353. iT - 5.06e4T^{2} \)
41 \( 1 - 83.7iT - 6.89e4T^{2} \)
43 \( 1 - 535.T + 7.95e4T^{2} \)
47 \( 1 - 411.T + 1.03e5T^{2} \)
53 \( 1 - 210. iT - 1.48e5T^{2} \)
59 \( 1 - 103. iT - 2.05e5T^{2} \)
61 \( 1 + 425.T + 2.26e5T^{2} \)
67 \( 1 + 796.T + 3.00e5T^{2} \)
71 \( 1 - 175. iT - 3.57e5T^{2} \)
73 \( 1 - 1.03e3iT - 3.89e5T^{2} \)
79 \( 1 - 273. iT - 4.93e5T^{2} \)
83 \( 1 + 486. iT - 5.71e5T^{2} \)
89 \( 1 + 267. iT - 7.04e5T^{2} \)
97 \( 1 - 603. iT - 9.12e5T^{2} \)
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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

−12.02951522489350622799900752618, −11.41344133801992869015453444450, −10.77432885799286818035086544631, −9.063569334792307810126492895042, −8.245990625666270696696618095607, −7.59012605208698101257981295862, −5.67349008028262511423003141369, −4.20485807962325644974606527664, −2.93701128803171164962742106483, −1.24477235052606891534492672562, 0.56546938713491209772727603313, 3.80457301355176768299420575172, 4.52594278322870194913023956846, 5.94188402519180887333109082556, 7.38415756974457601430411785993, 8.132291855909239229252454238132, 8.905528310148029483248876209598, 10.60233435197859908404748089159, 10.87237117044954111981173408662, 12.40037710819422341574064047341

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