Properties

Label 2-168-7.2-c3-0-8
Degree $2$
Conductor $168$
Sign $-0.175 + 0.984i$
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  + (−1.5 − 2.59i)3-s + (−9.47 + 16.4i)5-s + (12.8 − 13.3i)7-s + (−4.5 + 7.79i)9-s + (−27.3 − 47.3i)11-s + 62.0·13-s + 56.8·15-s + (−61.2 − 106. i)17-s + (6.25 − 10.8i)19-s + (−53.9 − 13.1i)21-s + (37.2 − 64.4i)23-s + (−117. − 203. i)25-s + 27·27-s − 232.·29-s + (−5.18 − 8.97i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.847 + 1.46i)5-s + (0.691 − 0.722i)7-s + (−0.166 + 0.288i)9-s + (−0.750 − 1.29i)11-s + 1.32·13-s + 0.978·15-s + (−0.873 − 1.51i)17-s + (0.0755 − 0.130i)19-s + (−0.560 − 0.137i)21-s + (0.337 − 0.584i)23-s + (−0.937 − 1.62i)25-s + 0.192·27-s − 1.48·29-s + (−0.0300 − 0.0520i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.175 + 0.984i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ -0.175 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.620934 - 0.741352i\)
\(L(\frac12)\) \(\approx\) \(0.620934 - 0.741352i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 + (-12.8 + 13.3i)T \)
good5 \( 1 + (9.47 - 16.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (27.3 + 47.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 62.0T + 2.19e3T^{2} \)
17 \( 1 + (61.2 + 106. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-6.25 + 10.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-37.2 + 64.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 232.T + 2.43e4T^{2} \)
31 \( 1 + (5.18 + 8.97i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-122. + 213. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 238.T + 6.89e4T^{2} \)
43 \( 1 + 92.9T + 7.95e4T^{2} \)
47 \( 1 + (-242. + 420. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-189. - 327. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-91.3 - 158. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (198. - 343. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-130. - 226. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 874.T + 3.57e5T^{2} \)
73 \( 1 + (76.2 + 131. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (286. - 496. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 317.T + 5.71e5T^{2} \)
89 \( 1 + (-47.5 + 82.2i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.60e3T + 9.12e5T^{2} \)
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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

−11.59247149858468889252833483970, −11.06865184649860388951238526040, −10.66880889553210943785173141696, −8.718502763199744147126787275771, −7.62185629625713816015530341358, −6.98424429722705676126952196737, −5.74333340403877349469782524656, −4.00455925916610848761935904876, −2.74298192852873173424704292698, −0.48148941959570695218262568299, 1.60195402923005614836986799945, 3.99919149971048796991793690535, 4.82713014809901132334423929791, 5.86122206868806473572176012404, 7.75763791523220469491928184500, 8.548069614648120679031211198382, 9.362366581762346604689184795692, 10.82662729537695756453911667810, 11.60413890179999848695336035557, 12.64026365794903157314141766087

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