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Properties

Label 2-168-7.2-c9-0-0
Degree $2$
Conductor $168$
Sign $-0.580 + 0.814i$
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 + (−547. + 948. i)5-s + (−5.93e3 + 2.26e3i)7-s + (−3.28e3 + 5.68e3i)9-s + (1.79e4 + 3.11e4i)11-s − 5.20e4·13-s + 8.87e4·15-s + (7.01e4 + 1.21e5i)17-s + (−5.07e5 + 8.78e5i)19-s + (3.99e5 + 3.24e5i)21-s + (1.34e5 − 2.32e5i)23-s + (3.76e5 + 6.51e5i)25-s + 5.31e5·27-s − 8.08e4·29-s + (4.97e5 + 8.61e5i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.500i)3-s + (−0.391 + 0.678i)5-s + (−0.934 + 0.356i)7-s + (−0.166 + 0.288i)9-s + (0.370 + 0.642i)11-s − 0.505·13-s + 0.452·15-s + (0.203 + 0.352i)17-s + (−0.892 + 1.54i)19-s + (0.447 + 0.364i)21-s + (0.100 − 0.173i)23-s + (0.192 + 0.333i)25-s + 0.192·27-s − 0.0212·29-s + (0.0967 + 0.167i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.580 + 0.814i$
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: \((2,\ 168,\ (\ :9/2),\ -0.580 + 0.814i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.06226124643\)
\(L(\frac12)\) \(\approx\) \(0.06226124643\)
\(L(\frac{11}{2})\) 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 \( 1 + (5.93e3 - 2.26e3i)T \)
good5 \( 1 + (547. - 948. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (-1.79e4 - 3.11e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + 5.20e4T + 1.06e10T^{2} \)
17 \( 1 + (-7.01e4 - 1.21e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (5.07e5 - 8.78e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-1.34e5 + 2.32e5i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + 8.08e4T + 1.45e13T^{2} \)
31 \( 1 + (-4.97e5 - 8.61e5i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (3.81e6 - 6.61e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 + 1.59e7T + 3.27e14T^{2} \)
43 \( 1 - 1.81e7T + 5.02e14T^{2} \)
47 \( 1 + (-2.19e7 + 3.80e7i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (-2.72e7 - 4.71e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (5.81e6 + 1.00e7i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (7.30e7 - 1.26e8i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (4.76e7 + 8.24e7i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 + 1.96e8T + 4.58e16T^{2} \)
73 \( 1 + (7.17e7 + 1.24e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-2.66e7 + 4.61e7i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + 2.96e8T + 1.86e17T^{2} \)
89 \( 1 + (3.90e8 - 6.76e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 + 8.87e7T + 7.60e17T^{2} \)
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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

−12.07673236028589689654756444420, −10.68308721907911005104610578418, −9.930031349997975924226833125606, −8.651028301517388184454649001804, −7.41911775264848464652868888371, −6.64053152116662426332163673253, −5.68509755605403886371555884860, −4.07806478277881389497846136366, −2.90705739090785696355487733985, −1.65718112092786995064571608595, 0.02004757316239553707049951967, 0.78082070599821488136722661961, 2.76222591390441526513280297256, 3.97605996082959607315008262348, 4.91057738158006489916413160814, 6.18897554219466650618670640369, 7.22896983867905940969466172195, 8.669683321584096738370926203608, 9.369795018360029351365398501036, 10.45927240684248376044787911399

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