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Properties

Label 2-168-7.3-c4-0-6
Degree $2$
Conductor $168$
Sign $0.638 + 0.769i$
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 + (−31.0 + 17.9i)5-s + (−41.3 + 26.2i)7-s + (13.5 + 23.3i)9-s + (21.0 − 36.5i)11-s − 22.7i·13-s + 186.·15-s + (111. + 64.3i)17-s + (338. − 195. i)19-s + (254. − 10.6i)21-s + (−331. − 574. i)23-s + (329. − 570. i)25-s − 140. i·27-s − 604.·29-s + (1.10e3 + 635. i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−1.24 + 0.716i)5-s + (−0.844 + 0.535i)7-s + (0.166 + 0.288i)9-s + (0.174 − 0.301i)11-s − 0.134i·13-s + 0.827·15-s + (0.385 + 0.222i)17-s + (0.938 − 0.541i)19-s + (0.576 − 0.0241i)21-s + (−0.626 − 1.08i)23-s + (0.527 − 0.913i)25-s − 0.192i·27-s − 0.718·29-s + (1.14 + 0.661i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(17.3661\)
Root analytic conductor: \(4.16727\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :2),\ 0.638 + 0.769i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.7619555565\)
\(L(\frac12)\) \(\approx\) \(0.7619555565\)
\(L(3)\) 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 \( 1 + (41.3 - 26.2i)T \)
good5 \( 1 + (31.0 - 17.9i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (-21.0 + 36.5i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 22.7iT - 2.85e4T^{2} \)
17 \( 1 + (-111. - 64.3i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-338. + 195. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (331. + 574. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 604.T + 7.07e5T^{2} \)
31 \( 1 + (-1.10e3 - 635. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (1.06e3 + 1.85e3i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 244. iT - 2.82e6T^{2} \)
43 \( 1 - 3.48e3T + 3.41e6T^{2} \)
47 \( 1 + (-114. + 66.1i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-481. + 833. i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-5.54e3 - 3.20e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (524. - 302. i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-1.59e3 + 2.76e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 6.48e3T + 2.54e7T^{2} \)
73 \( 1 + (-526. - 304. i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-596. - 1.03e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 4.57e3iT - 4.74e7T^{2} \)
89 \( 1 + (9.45e3 - 5.46e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 9.24e3iT - 8.85e7T^{2} \)
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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

−11.99114395633572204543842576123, −11.13254298943547762542237069541, −10.16279711966849421898029647063, −8.810376697639588450883867727072, −7.61347150970958472208918699086, −6.74783297794313144856827567241, −5.64005010935834659579413090283, −3.97560325182182917392623017877, −2.79914241190998931366607112721, −0.43788530602779012103677409357, 0.892385152196412788346291630491, 3.50030044325120931865963635613, 4.36497760417418593340106844988, 5.69710025173380811257671494946, 7.11980102711111245195144524087, 7.989558375191377206257228842778, 9.371602327400640407772869861928, 10.16712240625025203374096517740, 11.54804352869362569714564865478, 12.03748970269355850906477820577

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