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Properties

Label 2-162-9.4-c7-0-20
Degree $2$
Conductor $162$
Sign $0.642 + 0.766i$
Analytic cond. $50.6063$
Root an. cond. $7.11381$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 + 6.92i)2-s + (−31.9 + 55.4i)4-s + (−140. + 242. i)5-s + (82.1 + 142. i)7-s − 511.·8-s − 2.24e3·10-s + (−1.01e3 − 1.76e3i)11-s + (839. − 1.45e3i)13-s + (−657. + 1.13e3i)14-s + (−2.04e3 − 3.54e3i)16-s − 3.16e4·17-s − 1.26e4·19-s + (−8.97e3 − 1.55e4i)20-s + (8.15e3 − 1.41e4i)22-s + (2.47e4 − 4.28e4i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.501 + 0.868i)5-s + (0.0905 + 0.156i)7-s − 0.353·8-s − 0.709·10-s + (−0.230 − 0.399i)11-s + (0.105 − 0.183i)13-s + (−0.0640 + 0.110i)14-s + (−0.125 − 0.216i)16-s − 1.56·17-s − 0.422·19-s + (−0.250 − 0.434i)20-s + (0.163 − 0.282i)22-s + (0.423 − 0.734i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(50.6063\)
Root analytic conductor: \(7.11381\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :7/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.7522260178\)
\(L(\frac12)\) \(\approx\) \(0.7522260178\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 6.92i)T \)
3 \( 1 \)
good5 \( 1 + (140. - 242. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-82.1 - 142. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (1.01e3 + 1.76e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (-839. + 1.45e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + 3.16e4T + 4.10e8T^{2} \)
19 \( 1 + 1.26e4T + 8.93e8T^{2} \)
23 \( 1 + (-2.47e4 + 4.28e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (3.77e3 + 6.54e3i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (7.80e4 - 1.35e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 - 5.41e5T + 9.49e10T^{2} \)
41 \( 1 + (-2.68e5 + 4.65e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (1.00e5 + 1.73e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (2.05e5 + 3.55e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + 1.36e6T + 1.17e12T^{2} \)
59 \( 1 + (-3.99e5 + 6.91e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (2.84e5 + 4.93e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-2.40e6 + 4.16e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 2.45e6T + 9.09e12T^{2} \)
73 \( 1 - 1.60e6T + 1.10e13T^{2} \)
79 \( 1 + (2.79e6 + 4.84e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-4.91e6 - 8.51e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 - 1.17e5T + 4.42e13T^{2} \)
97 \( 1 + (-3.89e6 - 6.74e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−11.27856046762426009518608264904, −10.69702826595291026357954983290, −9.114586890988200889597645277416, −8.155946077936596754567930182838, −7.04598709820499543121941941252, −6.25645082414862481367745455556, −4.86982065545701607991043717087, −3.65247478255503966023656173234, −2.43849562915288251758386663072, −0.19180994383781480896047004425, 1.12164460266218258302419113557, 2.49428512813223368551869633917, 4.13961924688724109979990293844, 4.75718412311501318500343169204, 6.20564792093702131606263510822, 7.64596237776494907574527163591, 8.786935472804443317399783452182, 9.650233473653970029272149123892, 10.99978156115803792020124251684, 11.60638019925964351050311500717

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