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Properties

Label 2-168-56.27-c3-0-47
Degree $2$
Conductor $168$
Sign $-0.581 - 0.813i$
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  + (1.95 − 2.04i)2-s − 3i·3-s + (−0.332 − 7.99i)4-s − 14.3·5-s + (−6.12 − 5.87i)6-s + (−4.16 + 18.0i)7-s + (−16.9 − 14.9i)8-s − 9·9-s + (−28.1 + 29.3i)10-s + 20.6·11-s + (−23.9 + 0.996i)12-s − 55.3·13-s + (28.6 + 43.8i)14-s + 43.1i·15-s + (−63.7 + 5.31i)16-s + 35.0i·17-s + ⋯
L(s)  = 1  + (0.692 − 0.721i)2-s − 0.577i·3-s + (−0.0415 − 0.999i)4-s − 1.28·5-s + (−0.416 − 0.399i)6-s + (−0.224 + 0.974i)7-s + (−0.749 − 0.661i)8-s − 0.333·9-s + (−0.890 + 0.928i)10-s + 0.566·11-s + (−0.576 + 0.0239i)12-s − 1.18·13-s + (0.547 + 0.836i)14-s + 0.742i·15-s + (−0.996 + 0.0829i)16-s + 0.500i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.581 - 0.813i$
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: \((2,\ 168,\ (\ :3/2),\ -0.581 - 0.813i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.208966 + 0.406441i\)
\(L(\frac12)\) \(\approx\) \(0.208966 + 0.406441i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.95 + 2.04i)T \)
3 \( 1 + 3iT \)
7 \( 1 + (4.16 - 18.0i)T \)
good5 \( 1 + 14.3T + 125T^{2} \)
11 \( 1 - 20.6T + 1.33e3T^{2} \)
13 \( 1 + 55.3T + 2.19e3T^{2} \)
17 \( 1 - 35.0iT - 4.91e3T^{2} \)
19 \( 1 + 45.4iT - 6.85e3T^{2} \)
23 \( 1 + 14.3iT - 1.21e4T^{2} \)
29 \( 1 + 223. iT - 2.43e4T^{2} \)
31 \( 1 + 186.T + 2.97e4T^{2} \)
37 \( 1 + 57.0iT - 5.06e4T^{2} \)
41 \( 1 + 29.1iT - 6.89e4T^{2} \)
43 \( 1 + 461.T + 7.95e4T^{2} \)
47 \( 1 - 101.T + 1.03e5T^{2} \)
53 \( 1 - 342. iT - 1.48e5T^{2} \)
59 \( 1 + 621. iT - 2.05e5T^{2} \)
61 \( 1 - 877.T + 2.26e5T^{2} \)
67 \( 1 + 633.T + 3.00e5T^{2} \)
71 \( 1 + 28.6iT - 3.57e5T^{2} \)
73 \( 1 + 575. iT - 3.89e5T^{2} \)
79 \( 1 + 596. iT - 4.93e5T^{2} \)
83 \( 1 + 1.02e3iT - 5.71e5T^{2} \)
89 \( 1 - 968. iT - 7.04e5T^{2} \)
97 \( 1 - 1.57e3iT - 9.12e5T^{2} \)
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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

−11.92053514056689498462947583995, −11.19477297027050434091341146552, −9.750330907803068936622788164276, −8.652446303273910310298009271433, −7.36795287000230222615719485375, −6.17817570341956264696627798686, −4.84831800176367063597486322728, −3.54955670140553960378885235620, −2.20420790378863945067616704619, −0.16080569470567079748565439012, 3.33778361775441125565942550866, 4.14313694710982616960524311207, 5.15924637606295904478203992478, 6.89370559628449480138302834861, 7.51873640853693207463882497713, 8.653416042908866717127991945329, 9.954960405784669011243274247781, 11.28197787282277086793647502588, 12.02008405787398228894776130870, 12.99793889844094483712380785659

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