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Properties

Label 2-162-9.7-c5-0-13
Degree $2$
Conductor $162$
Sign $0.766 + 0.642i$
Analytic cond. $25.9821$
Root an. cond. $5.09727$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)2-s + (−7.99 − 13.8i)4-s + (42 + 72.7i)5-s + (96.5 − 167. i)7-s + 63.9·8-s − 336·10-s + (174 − 301. i)11-s + (−422.5 − 731. i)13-s + (386 + 668. i)14-s + (−128 + 221. i)16-s − 1.69e3·17-s − 79·19-s + (672 − 1.16e3i)20-s + (696 + 1.20e3i)22-s + (−282 − 488. i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.751 + 1.30i)5-s + (0.744 − 1.28i)7-s + 0.353·8-s − 1.06·10-s + (0.433 − 0.750i)11-s + (−0.693 − 1.20i)13-s + (0.526 + 0.911i)14-s + (−0.125 + 0.216i)16-s − 1.41·17-s − 0.0502·19-s + (0.375 − 0.650i)20-s + (0.306 + 0.531i)22-s + (−0.111 − 0.192i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(25.9821\)
Root analytic conductor: \(5.09727\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :5/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.502577135\)
\(L(\frac12)\) \(\approx\) \(1.502577135\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 - 3.46i)T \)
3 \( 1 \)
good5 \( 1 + (-42 - 72.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-96.5 + 167. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-174 + 301. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (422.5 + 731. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 1.69e3T + 1.41e6T^{2} \)
19 \( 1 + 79T + 2.47e6T^{2} \)
23 \( 1 + (282 + 488. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-3.21e3 + 5.57e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (2.47e3 + 4.27e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 3.80e3T + 6.93e7T^{2} \)
41 \( 1 + (6.24e3 + 1.08e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-2.46e3 + 4.27e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-4.06e3 + 7.03e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 3.31e4T + 4.18e8T^{2} \)
59 \( 1 + (-2.12e4 - 3.67e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-8.91e3 + 1.54e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-3.38e4 - 5.86e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 2.81e4T + 1.80e9T^{2} \)
73 \( 1 + 1.39e4T + 2.07e9T^{2} \)
79 \( 1 + (-4.19e4 + 7.27e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (1.66e4 - 2.89e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 7.78e4T + 5.58e9T^{2} \)
97 \( 1 + (-1.04e3 + 1.80e3i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−11.46326366265977404668892553948, −10.53914852831916839699276723988, −10.11178330095123158429771520958, −8.612975127062494389332868233632, −7.44992788829986564092597309165, −6.72498161005839064641572487743, −5.60062560786406639146654172901, −4.05091563182818141220053281248, −2.36170283160482396916777249695, −0.55108011856373619595216830498, 1.54280496346752375803431624158, 2.22056691979253182518084928382, 4.51724769836258279719697298156, 5.19485153179162738957525793120, 6.79628945425255140066514759818, 8.543864177320033134863076134925, 8.993232413470233434922444012829, 9.756926164438218341017643151664, 11.26315466326530653463021020761, 12.15490745415653075997527975173

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