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Properties

Label 2-168-56.27-c3-0-43
Degree $2$
Conductor $168$
Sign $-0.996 + 0.0787i$
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  + (2.14 − 1.84i)2-s + 3i·3-s + (1.17 − 7.91i)4-s − 7.57·5-s + (5.54 + 6.42i)6-s + (−16.3 − 8.64i)7-s + (−12.1 − 19.1i)8-s − 9·9-s + (−16.2 + 13.9i)10-s − 13.4·11-s + (23.7 + 3.52i)12-s − 27.1·13-s + (−51.0 + 11.7i)14-s − 22.7i·15-s + (−61.2 − 18.5i)16-s − 35.4i·17-s + ⋯
L(s)  = 1  + (0.757 − 0.653i)2-s + 0.577i·3-s + (0.146 − 0.989i)4-s − 0.677·5-s + (0.377 + 0.437i)6-s + (−0.884 − 0.466i)7-s + (−0.534 − 0.844i)8-s − 0.333·9-s + (−0.513 + 0.442i)10-s − 0.367·11-s + (0.571 + 0.0847i)12-s − 0.579·13-s + (−0.974 + 0.224i)14-s − 0.391i·15-s + (−0.956 − 0.290i)16-s − 0.505i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0787i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.996 + 0.0787i$
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: \((2,\ 168,\ (\ :3/2),\ -0.996 + 0.0787i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0332084 - 0.842519i\)
\(L(\frac12)\) \(\approx\) \(0.0332084 - 0.842519i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.14 + 1.84i)T \)
3 \( 1 - 3iT \)
7 \( 1 + (16.3 + 8.64i)T \)
good5 \( 1 + 7.57T + 125T^{2} \)
11 \( 1 + 13.4T + 1.33e3T^{2} \)
13 \( 1 + 27.1T + 2.19e3T^{2} \)
17 \( 1 + 35.4iT - 4.91e3T^{2} \)
19 \( 1 + 3.51iT - 6.85e3T^{2} \)
23 \( 1 + 84.3iT - 1.21e4T^{2} \)
29 \( 1 - 32.8iT - 2.43e4T^{2} \)
31 \( 1 - 82.5T + 2.97e4T^{2} \)
37 \( 1 - 71.5iT - 5.06e4T^{2} \)
41 \( 1 + 236. iT - 6.89e4T^{2} \)
43 \( 1 - 539.T + 7.95e4T^{2} \)
47 \( 1 + 375.T + 1.03e5T^{2} \)
53 \( 1 - 407. iT - 1.48e5T^{2} \)
59 \( 1 + 424. iT - 2.05e5T^{2} \)
61 \( 1 + 246.T + 2.26e5T^{2} \)
67 \( 1 - 527.T + 3.00e5T^{2} \)
71 \( 1 + 780. iT - 3.57e5T^{2} \)
73 \( 1 - 353. iT - 3.89e5T^{2} \)
79 \( 1 + 522. iT - 4.93e5T^{2} \)
83 \( 1 + 1.32e3iT - 5.71e5T^{2} \)
89 \( 1 + 626. iT - 7.04e5T^{2} \)
97 \( 1 - 313. iT - 9.12e5T^{2} \)
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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

−11.92642761921045697116792296197, −10.86685622841063657255050430770, −10.08773711989384908400359250491, −9.167137880349901528231048927518, −7.51538236731843131205327488031, −6.25617500635825446393590156208, −4.88005069145590541863597051044, −3.87518852603549113485284356297, −2.75091982895330550539333883136, −0.28533154036469836428198314806, 2.63307491570714345989375842176, 3.90680410704846573425606545505, 5.43071501273006431290662526821, 6.45511356327898445529729305522, 7.49138680217879239058605112530, 8.309494593266263233226939569386, 9.617171144619887892576525882704, 11.26972477655018701457094571499, 12.17804343206530355655767192920, 12.81123575627819253918854052169

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