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Properties

Label 2-162-9.4-c3-0-5
Degree $2$
Conductor $162$
Sign $0.939 + 0.342i$
Analytic cond. $9.55830$
Root an. cond. $3.09165$
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 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (3 − 5.19i)5-s + (8 + 13.8i)7-s + 7.99·8-s − 12·10-s + (6 + 10.3i)11-s + (−19 + 32.9i)13-s + (15.9 − 27.7i)14-s + (−8 − 13.8i)16-s + 126·17-s + 20·19-s + (12.0 + 20.7i)20-s + (12 − 20.7i)22-s + (84 − 145. i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.268 − 0.464i)5-s + (0.431 + 0.748i)7-s + 0.353·8-s − 0.379·10-s + (0.164 + 0.284i)11-s + (−0.405 + 0.702i)13-s + (0.305 − 0.529i)14-s + (−0.125 − 0.216i)16-s + 1.79·17-s + 0.241·19-s + (0.134 + 0.232i)20-s + (0.116 − 0.201i)22-s + (0.761 − 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(9.55830\)
Root analytic conductor: \(3.09165\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.52372 - 0.268673i\)
\(L(\frac12)\) \(\approx\) \(1.52372 - 0.268673i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (-3 + 5.19i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-8 - 13.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-6 - 10.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (19 - 32.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 126T + 4.91e3T^{2} \)
19 \( 1 - 20T + 6.85e3T^{2} \)
23 \( 1 + (-84 + 145. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-15 - 25.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-44 + 76.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 254T + 5.06e4T^{2} \)
41 \( 1 + (-21 + 36.3i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-26 - 45.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (48 + 83.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 198T + 1.48e5T^{2} \)
59 \( 1 + (330 - 571. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-269 - 465. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (442 - 765. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 792T + 3.57e5T^{2} \)
73 \( 1 - 218T + 3.89e5T^{2} \)
79 \( 1 + (-260 - 450. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (246 + 426. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 810T + 7.04e5T^{2} \)
97 \( 1 + (577 + 999. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.20839397221509747726704389949, −11.51037612533864586149893724782, −10.22591675239314802363510881460, −9.324928372625985479939020076123, −8.489242657620828730449982424819, −7.28963822072086638822669320879, −5.65329438305695635211099530227, −4.50686071330053995325001594584, −2.75760071290381303872345961737, −1.27481194623403663145947945777, 1.06071706954334300248305886668, 3.25969045505317201722044621140, 4.96916493000047226129319386859, 6.09384407451649237276719747587, 7.39304984983297156048979627129, 8.002542318996537816098215346162, 9.499692523440939874521279418580, 10.28851763362464824538447798509, 11.22510497842326500104040318332, 12.52672256973751990783629538386

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