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Properties

Label 2-168-7.4-c9-0-24
Degree $2$
Conductor $168$
Sign $0.958 + 0.285i$
Analytic cond. $86.5260$
Root an. cond. $9.30193$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−40.5 + 70.1i)3-s + (186. + 323. i)5-s + (6.07e3 − 1.86e3i)7-s + (−3.28e3 − 5.68e3i)9-s + (−1.81e3 + 3.14e3i)11-s + 1.98e5·13-s − 3.02e4·15-s + (−6.03e4 + 1.04e5i)17-s + (−1.81e5 − 3.13e5i)19-s + (−1.15e5 + 5.01e5i)21-s + (3.04e4 + 5.27e4i)23-s + (9.06e5 − 1.57e6i)25-s + 5.31e5·27-s − 1.46e6·29-s + (2.44e6 − 4.23e6i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.500i)3-s + (0.133 + 0.231i)5-s + (0.956 − 0.293i)7-s + (−0.166 − 0.288i)9-s + (−0.0374 + 0.0648i)11-s + 1.92·13-s − 0.154·15-s + (−0.175 + 0.303i)17-s + (−0.318 − 0.551i)19-s + (−0.129 + 0.562i)21-s + (0.0226 + 0.0393i)23-s + (0.464 − 0.804i)25-s + 0.192·27-s − 0.385·29-s + (0.475 − 0.823i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.958 + 0.285i$
Analytic conductor: \(86.5260\)
Root analytic conductor: \(9.30193\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :9/2),\ 0.958 + 0.285i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.386839168\)
\(L(\frac12)\) \(\approx\) \(2.386839168\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (40.5 - 70.1i)T \)
7 \( 1 + (-6.07e3 + 1.86e3i)T \)
good5 \( 1 + (-186. - 323. i)T + (-9.76e5 + 1.69e6i)T^{2} \)
11 \( 1 + (1.81e3 - 3.14e3i)T + (-1.17e9 - 2.04e9i)T^{2} \)
13 \( 1 - 1.98e5T + 1.06e10T^{2} \)
17 \( 1 + (6.03e4 - 1.04e5i)T + (-5.92e10 - 1.02e11i)T^{2} \)
19 \( 1 + (1.81e5 + 3.13e5i)T + (-1.61e11 + 2.79e11i)T^{2} \)
23 \( 1 + (-3.04e4 - 5.27e4i)T + (-9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 + 1.46e6T + 1.45e13T^{2} \)
31 \( 1 + (-2.44e6 + 4.23e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (7.85e6 + 1.36e7i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 + 2.52e7T + 3.27e14T^{2} \)
43 \( 1 + 1.07e7T + 5.02e14T^{2} \)
47 \( 1 + (-7.55e6 - 1.30e7i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + (1.29e6 - 2.24e6i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 + (-7.47e7 + 1.29e8i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (-7.16e7 - 1.24e8i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (-3.68e7 + 6.38e7i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + 2.94e8T + 4.58e16T^{2} \)
73 \( 1 + (-5.41e7 + 9.38e7i)T + (-2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (6.08e7 + 1.05e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 - 5.78e8T + 1.86e17T^{2} \)
89 \( 1 + (-2.47e8 - 4.28e8i)T + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 - 3.41e8T + 7.60e17T^{2} \)
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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

−10.96705208597812274430423137277, −10.34993740708748938517687541391, −8.929108531581899476601796573722, −8.175291260750569163337070128816, −6.75302161566346737735967704916, −5.73191259453729420548513453243, −4.53154855899551145661213253087, −3.56161637704757904228663716710, −1.92448763052573633450894048178, −0.66073875151626859441611579003, 1.04932227321748876248540641124, 1.79860487841629529250979586882, 3.44738422557805204462103948423, 4.89181598592669706264297087263, 5.85665736381691131392297529540, 6.92471257209630631074354432847, 8.291630714643957679545866082295, 8.754795386606267392991309795899, 10.37652166412256880344563869710, 11.26922015979836998887968942053

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