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Properties

Label 2-168-7.2-c9-0-28
Degree $2$
Conductor $168$
Sign $-0.987 + 0.156i$
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 + (−519. + 899. i)5-s + (−5.77e3 − 2.64e3i)7-s + (−3.28e3 + 5.68e3i)9-s + (−2.47e4 − 4.28e4i)11-s + 1.34e5·13-s + 8.40e4·15-s + (2.53e5 + 4.38e5i)17-s + (4.75e5 − 8.23e5i)19-s + (4.79e4 + 5.12e5i)21-s + (−4.36e5 + 7.55e5i)23-s + (4.37e5 + 7.58e5i)25-s + 5.31e5·27-s − 1.12e6·29-s + (−8.85e5 − 1.53e6i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.500i)3-s + (−0.371 + 0.643i)5-s + (−0.908 − 0.417i)7-s + (−0.166 + 0.288i)9-s + (−0.509 − 0.881i)11-s + 1.30·13-s + 0.428·15-s + (0.735 + 1.27i)17-s + (0.836 − 1.44i)19-s + (0.0538 + 0.574i)21-s + (−0.324 + 0.562i)23-s + (0.224 + 0.388i)25-s + 0.192·27-s − 0.295·29-s + (−0.172 − 0.298i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.987 + 0.156i$
Analytic conductor: \(86.5260\)
Root analytic conductor: \(9.30193\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :9/2),\ -0.987 + 0.156i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.4216367606\)
\(L(\frac12)\) \(\approx\) \(0.4216367606\)
\(L(\frac{11}{2})\) 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 \( 1 + (5.77e3 + 2.64e3i)T \)
good5 \( 1 + (519. - 899. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (2.47e4 + 4.28e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 - 1.34e5T + 1.06e10T^{2} \)
17 \( 1 + (-2.53e5 - 4.38e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-4.75e5 + 8.23e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (4.36e5 - 7.55e5i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + 1.12e6T + 1.45e13T^{2} \)
31 \( 1 + (8.85e5 + 1.53e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (-2.51e6 + 4.35e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 2.26e7T + 3.27e14T^{2} \)
43 \( 1 + 2.75e7T + 5.02e14T^{2} \)
47 \( 1 + (8.13e6 - 1.40e7i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (1.96e7 + 3.39e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (7.10e7 + 1.23e8i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (-8.11e7 + 1.40e8i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (-8.79e7 - 1.52e8i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 + 3.10e8T + 4.58e16T^{2} \)
73 \( 1 + (-7.28e7 - 1.26e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (7.60e7 - 1.31e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + 7.68e8T + 1.86e17T^{2} \)
89 \( 1 + (-3.06e7 + 5.30e7i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 + 1.09e9T + 7.60e17T^{2} \)
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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

−10.95864718270253357919422836427, −9.723402573459513577038327498725, −8.414023144629132146903225209269, −7.41405533864587502736488029646, −6.43294460648483490143784676781, −5.59426957968971775099537417215, −3.72890295528763255768209061785, −3.00033970800767819729553528319, −1.23052069254993288544739681967, −0.11734970312909151353673541092, 1.16244653987652446602746957468, 2.92846256185562363584056524612, 4.03063730780402489229226809125, 5.21830613101496555797508178974, 6.15013129796577385210711610229, 7.53302808390434093792624047470, 8.667073899676713251796117547815, 9.657329304693211799474383633080, 10.38642117667307265994787687224, 11.78766540079784205017861179088

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