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Properties

Label 2-168-7.4-c3-0-9
Degree $2$
Conductor $168$
Sign $-0.514 + 0.857i$
Analytic cond. $9.91232$
Root an. cond. $3.14838$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 2.59i)3-s + (−0.0642 − 0.111i)5-s + (−0.866 − 18.4i)7-s + (−4.5 − 7.79i)9-s + (−27.0 + 46.7i)11-s − 50.2·13-s + 0.385·15-s + (65.7 − 113. i)17-s + (−45.7 − 79.2i)19-s + (49.3 + 25.4i)21-s + (−89.7 − 155. i)23-s + (62.4 − 108. i)25-s + 27·27-s − 69.8·29-s + (−163. + 283. i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.00575 − 0.00996i)5-s + (−0.0467 − 0.998i)7-s + (−0.166 − 0.288i)9-s + (−0.740 + 1.28i)11-s − 1.07·13-s + 0.00664·15-s + (0.937 − 1.62i)17-s + (−0.552 − 0.956i)19-s + (0.512 + 0.264i)21-s + (−0.813 − 1.40i)23-s + (0.499 − 0.865i)25-s + 0.192·27-s − 0.447·29-s + (−0.946 + 1.63i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.514 + 0.857i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ -0.514 + 0.857i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.292775 - 0.516954i\)
\(L(\frac12)\) \(\approx\) \(0.292775 - 0.516954i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 + (0.866 + 18.4i)T \)
good5 \( 1 + (0.0642 + 0.111i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (27.0 - 46.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 50.2T + 2.19e3T^{2} \)
17 \( 1 + (-65.7 + 113. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (45.7 + 79.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (89.7 + 155. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 69.8T + 2.43e4T^{2} \)
31 \( 1 + (163. - 283. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (150. + 261. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 296.T + 6.89e4T^{2} \)
43 \( 1 + 144.T + 7.95e4T^{2} \)
47 \( 1 + (-180. - 311. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (0.917 - 1.58i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-26.6 + 46.1i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-54.0 - 93.6i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (421. - 729. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 241.T + 3.57e5T^{2} \)
73 \( 1 + (-103. + 179. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (279. + 484. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 986.T + 5.71e5T^{2} \)
89 \( 1 + (-221. - 383. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 740.T + 9.12e5T^{2} \)
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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

−12.12371987719857662479646443061, −10.70700854002841857554314338645, −10.16146239072259905580990599122, −9.189018595676111605388929550550, −7.57272944656093441362407131213, −6.91185283562338134070212989347, −5.11289019483971107610220311166, −4.39716263435499457967613124075, −2.61950868230543554016296712773, −0.26902599466842819701376142815, 1.89933673808477707209655276833, 3.46653697037314657689393421771, 5.51873760625707719971575710861, 5.94914851902795565285532055474, 7.64761795829324442560145128202, 8.358624015464593098450976666330, 9.671654006516248423449680475763, 10.78600644501675476400764921326, 11.81362618268353471166998023775, 12.60155856026461834513831032761

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