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Properties

Label 2-168-7.4-c9-0-26
Degree $2$
Conductor $168$
Sign $0.0665 + 0.997i$
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 + (870. + 1.50e3i)5-s + (3.86e3 + 5.04e3i)7-s + (−3.28e3 − 5.68e3i)9-s + (−4.03e3 + 6.98e3i)11-s − 1.01e5·13-s − 1.40e5·15-s + (1.47e5 − 2.56e5i)17-s + (−1.86e5 − 3.23e5i)19-s + (−5.10e5 + 6.66e4i)21-s + (−1.19e6 − 2.07e6i)23-s + (−5.38e5 + 9.32e5i)25-s + 5.31e5·27-s − 5.40e6·29-s + (7.56e4 − 1.30e5i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.500i)3-s + (0.622 + 1.07i)5-s + (0.607 + 0.793i)7-s + (−0.166 − 0.288i)9-s + (−0.0830 + 0.143i)11-s − 0.986·13-s − 0.719·15-s + (0.429 − 0.744i)17-s + (−0.328 − 0.569i)19-s + (−0.572 + 0.0748i)21-s + (−0.892 − 1.54i)23-s + (−0.275 + 0.477i)25-s + 0.192·27-s − 1.41·29-s + (0.0147 − 0.0254i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0665 + 0.997i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0665 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.0665 + 0.997i$
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: \((2,\ 168,\ (\ :9/2),\ 0.0665 + 0.997i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.4350125805\)
\(L(\frac12)\) \(\approx\) \(0.4350125805\)
\(L(\frac{11}{2})\) 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 \( 1 + (-3.86e3 - 5.04e3i)T \)
good5 \( 1 + (-870. - 1.50e3i)T + (-9.76e5 + 1.69e6i)T^{2} \)
11 \( 1 + (4.03e3 - 6.98e3i)T + (-1.17e9 - 2.04e9i)T^{2} \)
13 \( 1 + 1.01e5T + 1.06e10T^{2} \)
17 \( 1 + (-1.47e5 + 2.56e5i)T + (-5.92e10 - 1.02e11i)T^{2} \)
19 \( 1 + (1.86e5 + 3.23e5i)T + (-1.61e11 + 2.79e11i)T^{2} \)
23 \( 1 + (1.19e6 + 2.07e6i)T + (-9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 + 5.40e6T + 1.45e13T^{2} \)
31 \( 1 + (-7.56e4 + 1.30e5i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (-7.89e4 - 1.36e5i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 - 1.33e7T + 3.27e14T^{2} \)
43 \( 1 + 1.32e7T + 5.02e14T^{2} \)
47 \( 1 + (-4.26e6 - 7.38e6i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + (-1.08e7 + 1.87e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 + (5.99e7 - 1.03e8i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (6.30e7 + 1.09e8i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (-3.07e7 + 5.32e7i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 - 2.83e7T + 4.58e16T^{2} \)
73 \( 1 + (-3.90e7 + 6.77e7i)T + (-2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (2.51e7 + 4.35e7i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 + 3.60e8T + 1.86e17T^{2} \)
89 \( 1 + (2.00e8 + 3.47e8i)T + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 - 1.53e9T + 7.60e17T^{2} \)
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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

−10.79428579434311265091030771238, −9.961107334374049224186891527535, −9.064744158805538028964658525596, −7.69364855539823229745019311715, −6.54444739477686099436649500115, −5.53998306291264420043732251030, −4.51810511941752038940742015643, −2.85520398188506372006579391647, −2.09122823937870253574814044823, −0.097565786015862908931861796751, 1.22020954007403775336662875382, 1.92374363646563268762019699434, 3.88857101776655156985382035094, 5.11468429528924845974116118542, 5.88661069324727453093144857926, 7.38993026747258678231144611423, 8.109853673399054040874809218743, 9.379047645714077053919912517148, 10.29383849702780348497468026404, 11.45583499293688957114743366591

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