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Properties

Label 2-168-7.2-c9-0-15
Degree $2$
Conductor $168$
Sign $0.998 + 0.0461i$
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 + (−915. + 1.58e3i)5-s + (5.12e3 + 3.75e3i)7-s + (−3.28e3 + 5.68e3i)9-s + (−2.88e4 − 5.00e4i)11-s − 1.49e5·13-s + 1.48e5·15-s + (3.23e4 + 5.60e4i)17-s + (2.88e5 − 4.99e5i)19-s + (5.62e4 − 5.11e5i)21-s + (5.83e5 − 1.01e6i)23-s + (−7.00e5 − 1.21e6i)25-s + 5.31e5·27-s − 3.62e6·29-s + (−1.45e6 − 2.51e6i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.500i)3-s + (−0.655 + 1.13i)5-s + (0.806 + 0.591i)7-s + (−0.166 + 0.288i)9-s + (−0.595 − 1.03i)11-s − 1.45·13-s + 0.756·15-s + (0.0939 + 0.162i)17-s + (0.507 − 0.879i)19-s + (0.0631 − 0.573i)21-s + (0.434 − 0.753i)23-s + (−0.358 − 0.621i)25-s + 0.192·27-s − 0.951·29-s + (−0.282 − 0.490i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0461i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.998 + 0.0461i$
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: \((2,\ 168,\ (\ :9/2),\ 0.998 + 0.0461i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.262463605\)
\(L(\frac12)\) \(\approx\) \(1.262463605\)
\(L(\frac{11}{2})\) 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 \( 1 + (-5.12e3 - 3.75e3i)T \)
good5 \( 1 + (915. - 1.58e3i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (2.88e4 + 5.00e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + 1.49e5T + 1.06e10T^{2} \)
17 \( 1 + (-3.23e4 - 5.60e4i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-2.88e5 + 4.99e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-5.83e5 + 1.01e6i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + 3.62e6T + 1.45e13T^{2} \)
31 \( 1 + (1.45e6 + 2.51e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (4.80e6 - 8.32e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 + 1.92e6T + 3.27e14T^{2} \)
43 \( 1 - 2.02e7T + 5.02e14T^{2} \)
47 \( 1 + (-2.37e6 + 4.12e6i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (-1.74e7 - 3.01e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (2.15e7 + 3.73e7i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (8.05e7 - 1.39e8i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (-1.41e8 - 2.44e8i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 - 1.95e8T + 4.58e16T^{2} \)
73 \( 1 + (1.14e8 + 1.97e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-1.24e8 + 2.15e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + 5.69e7T + 1.86e17T^{2} \)
89 \( 1 + (-5.58e8 + 9.67e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 - 1.19e9T + 7.60e17T^{2} \)
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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

−11.25121453684959811339585061199, −10.42139869479144680857823446874, −8.903879266465296066848646911246, −7.73840787685834635182366451763, −7.15999811233536317368029722329, −5.84458463312589864988634695849, −4.78847006694107518012569160762, −3.09505036136217720877818966151, −2.26966172582920415972743620722, −0.52365902593875906592553185013, 0.58955563015166438456754873525, 1.90933771551668717698772157789, 3.76984199836751088129617142456, 4.83358500841872675904772254589, 5.24446657788100459158824381044, 7.33478582327207096825416765783, 7.86596697145802780718382507748, 9.212797245990039122697286161351, 10.05597517357184996933915977222, 11.15334934447148962957561398395

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