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Properties

Label 2-168-168.101-c3-0-40
Degree $2$
Conductor $168$
Sign $0.855 - 0.517i$
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  + (0.342 + 2.80i)2-s + (−4.19 − 3.07i)3-s + (−7.76 + 1.92i)4-s + (3.83 − 2.21i)5-s + (7.18 − 12.8i)6-s + (−18.2 + 2.98i)7-s + (−8.05 − 21.1i)8-s + (8.13 + 25.7i)9-s + (7.53 + 10.0i)10-s + (4.57 − 7.91i)11-s + (38.4 + 15.7i)12-s + 69.6·13-s + (−14.6 − 50.2i)14-s + (−22.8 − 2.49i)15-s + (56.6 − 29.8i)16-s + (57.1 − 98.9i)17-s + ⋯
L(s)  = 1  + (0.121 + 0.992i)2-s + (−0.806 − 0.591i)3-s + (−0.970 + 0.240i)4-s + (0.343 − 0.198i)5-s + (0.489 − 0.872i)6-s + (−0.986 + 0.161i)7-s + (−0.356 − 0.934i)8-s + (0.301 + 0.953i)9-s + (0.238 + 0.316i)10-s + (0.125 − 0.217i)11-s + (0.925 + 0.379i)12-s + 1.48·13-s + (−0.279 − 0.960i)14-s + (−0.393 − 0.0430i)15-s + (0.884 − 0.466i)16-s + (0.814 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.855 - 0.517i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ 0.855 - 0.517i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.16316 + 0.324318i\)
\(L(\frac12)\) \(\approx\) \(1.16316 + 0.324318i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 - 2.80i)T \)
3 \( 1 + (4.19 + 3.07i)T \)
7 \( 1 + (18.2 - 2.98i)T \)
good5 \( 1 + (-3.83 + 2.21i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-4.57 + 7.91i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 69.6T + 2.19e3T^{2} \)
17 \( 1 + (-57.1 + 98.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-61.1 - 105. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-17.4 + 10.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 170.T + 2.43e4T^{2} \)
31 \( 1 + (21.2 + 12.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-128. + 74.4i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 232.T + 6.89e4T^{2} \)
43 \( 1 + 330. iT - 7.95e4T^{2} \)
47 \( 1 + (-118. - 205. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-165. + 287. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-378. - 218. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (261. + 453. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-149. - 86.5i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 792. iT - 3.57e5T^{2} \)
73 \( 1 + (-952. - 550. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (517. + 896. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 364. iT - 5.71e5T^{2} \)
89 \( 1 + (-475. - 823. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 538. iT - 9.12e5T^{2} \)
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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

−12.58012792812966714112075583602, −11.73311224193156444853419606131, −10.23227664086454155868383485461, −9.271452561677862265965484521421, −8.032915010585695823679533577196, −6.92700496589443359540497397536, −6.00247978826221544631806559149, −5.32051767468149968359604950465, −3.52996824371123803256623688568, −0.875377313781356747428927425030, 0.996861682154298623713166274759, 3.18014248362260469431010445982, 4.20235864460220519377893291422, 5.66732219247750140512900330189, 6.51933335973492975260109157879, 8.574898274425063267789912831921, 9.682330163027822907351699684525, 10.31186474208591940682709263416, 11.14755330396202161498757144101, 12.12903303072546155937007644398

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