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Properties

Label 2-168-7.5-c4-0-9
Degree $2$
Conductor $168$
Sign $0.592 + 0.805i$
Analytic cond. $17.3661$
Root an. cond. $4.16727$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 + 2.59i)3-s + (−6.72 − 3.88i)5-s + (−35.4 + 33.8i)7-s + (13.5 − 23.3i)9-s + (6.76 + 11.7i)11-s − 75.5i·13-s + 40.3·15-s + (117. − 68.1i)17-s + (36.5 + 21.1i)19-s + (71.7 − 244. i)21-s + (185. − 320. i)23-s + (−282. − 488. i)25-s + 140. i·27-s + 500.·29-s + (771. − 445. i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (−0.269 − 0.155i)5-s + (−0.723 + 0.689i)7-s + (0.166 − 0.288i)9-s + (0.0558 + 0.0967i)11-s − 0.447i·13-s + 0.179·15-s + (0.408 − 0.235i)17-s + (0.101 + 0.0584i)19-s + (0.162 − 0.553i)21-s + (0.349 − 0.605i)23-s + (−0.451 − 0.782i)25-s + 0.192i·27-s + 0.595·29-s + (0.803 − 0.463i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.592 + 0.805i$
Analytic conductor: \(17.3661\)
Root analytic conductor: \(4.16727\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :2),\ 0.592 + 0.805i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.037792687\)
\(L(\frac12)\) \(\approx\) \(1.037792687\)
\(L(3)\) 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 \( 1 + (4.5 - 2.59i)T \)
7 \( 1 + (35.4 - 33.8i)T \)
good5 \( 1 + (6.72 + 3.88i)T + (312.5 + 541. i)T^{2} \)
11 \( 1 + (-6.76 - 11.7i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 75.5iT - 2.85e4T^{2} \)
17 \( 1 + (-117. + 68.1i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-36.5 - 21.1i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-185. + 320. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 500.T + 7.07e5T^{2} \)
31 \( 1 + (-771. + 445. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-1.14e3 + 1.98e3i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + 1.03e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.93e3T + 3.41e6T^{2} \)
47 \( 1 + (-668. - 386. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (370. + 640. i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (2.54e3 - 1.46e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (3.01e3 + 1.74e3i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (844. + 1.46e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 2.28e3T + 2.54e7T^{2} \)
73 \( 1 + (1.22e3 - 705. i)T + (1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-365. + 633. i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.00e4iT - 4.74e7T^{2} \)
89 \( 1 + (-1.95e3 - 1.12e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 7.67e3iT - 8.85e7T^{2} \)
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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

−12.14863773794463351599728126076, −10.96446134358336774541678137352, −9.951093601218468481086297670075, −9.044728363208357411475095687806, −7.82257169350368546615894960442, −6.45354867130433661266437708282, −5.51909687378388057848374538766, −4.18941788541652455594936349218, −2.70812438077730878226458194226, −0.50100356807770728416289305691, 1.12763433888258349345336013624, 3.16484626253333221154593646576, 4.50757783266417663991292818742, 6.00080309138000258965678016975, 6.95138855165863853706422744504, 7.905147721556600444359990153469, 9.373593601850604818646752714896, 10.33053833187957355602980452355, 11.32117963781429217202464411632, 12.19693222892368241085058406846

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