Properties

Label 2-162-27.22-c3-0-0
Degree $2$
Conductor $162$
Sign $-0.670 - 0.741i$
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  + (1.53 − 1.28i)2-s + (0.694 − 3.93i)4-s + (−11.4 + 4.18i)5-s + (3.15 + 17.9i)7-s + (−4.00 − 6.92i)8-s + (−12.2 + 21.1i)10-s + (−63.8 − 23.2i)11-s + (−28.0 − 23.5i)13-s + (27.8 + 23.3i)14-s + (−15.0 − 5.47i)16-s + (−59.2 + 102. i)17-s + (51.9 + 89.9i)19-s + (8.49 + 48.2i)20-s + (−127. + 46.4i)22-s + (2.68 − 15.2i)23-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.0868 − 0.492i)4-s + (−1.02 + 0.374i)5-s + (0.170 + 0.966i)7-s + (−0.176 − 0.306i)8-s + (−0.386 + 0.670i)10-s + (−1.75 − 0.637i)11-s + (−0.599 − 0.502i)13-s + (0.531 + 0.446i)14-s + (−0.234 − 0.0855i)16-s + (−0.845 + 1.46i)17-s + (0.626 + 1.08i)19-s + (0.0950 + 0.538i)20-s + (−1.23 + 0.450i)22-s + (0.0242 − 0.137i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.670 - 0.741i$
Analytic conductor: \(9.55830\)
Root analytic conductor: \(3.09165\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3/2),\ -0.670 - 0.741i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.166615 + 0.375476i\)
\(L(\frac12)\) \(\approx\) \(0.166615 + 0.375476i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.53 + 1.28i)T \)
3 \( 1 \)
good5 \( 1 + (11.4 - 4.18i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (-3.15 - 17.9i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (63.8 + 23.2i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (28.0 + 23.5i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (59.2 - 102. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-51.9 - 89.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-2.68 + 15.2i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-113. + 95.1i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-15.3 + 86.8i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (47.5 - 82.3i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (188. + 158. i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (94.8 + 34.5i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-24.8 - 140. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + 213.T + 1.48e5T^{2} \)
59 \( 1 + (-480. + 174. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (75.9 + 431. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (680. + 570. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (202. - 350. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-37.3 - 64.6i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (469. - 393. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-165. + 138. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (-416. - 721. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.13e3 - 412. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−12.61187628190136755748305214231, −11.87012824928051600817396882796, −10.89365118763782276516024226307, −10.12337416606679218209379498219, −8.427561436351129778561381465108, −7.75775513174654055744451284681, −6.06830807036398786061116339043, −5.05542952087583451769502820645, −3.56783254273428420168623134326, −2.39903371176151743689617192541, 0.15065822496424915157315852772, 2.84224169577447337015869145162, 4.52493540465269923160739025113, 4.98569064018698999172781819067, 7.17012824859537516711997489222, 7.39396190715348664876593764660, 8.674366605262786181602002172231, 10.10484192044040853918697675260, 11.27012590378450213557977696727, 12.09470010352230886042533103044

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