Properties

Label 2-162-81.70-c1-0-8
Degree $2$
Conductor $162$
Sign $-0.998 - 0.0510i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.686 − 0.727i)2-s + (−1.68 − 0.388i)3-s + (−0.0581 − 0.998i)4-s + (−3.71 − 0.434i)5-s + (−1.44 + 0.961i)6-s + (−3.25 + 1.63i)7-s + (−0.766 − 0.642i)8-s + (2.69 + 1.31i)9-s + (−2.86 + 2.40i)10-s + (2.08 − 4.83i)11-s + (−0.289 + 1.70i)12-s + (−0.202 + 0.0480i)13-s + (−1.04 + 3.48i)14-s + (6.10 + 2.17i)15-s + (−0.993 + 0.116i)16-s + (−0.290 − 1.64i)17-s + ⋯
L(s)  = 1  + (0.485 − 0.514i)2-s + (−0.974 − 0.224i)3-s + (−0.0290 − 0.499i)4-s + (−1.66 − 0.194i)5-s + (−0.588 + 0.392i)6-s + (−1.22 + 0.617i)7-s + (−0.270 − 0.227i)8-s + (0.899 + 0.436i)9-s + (−0.906 + 0.760i)10-s + (0.628 − 1.45i)11-s + (−0.0835 + 0.492i)12-s + (−0.0562 + 0.0133i)13-s + (−0.278 + 0.931i)14-s + (1.57 + 0.562i)15-s + (−0.248 + 0.0290i)16-s + (−0.0703 − 0.399i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.998 - 0.0510i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ -0.998 - 0.0510i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00905092 + 0.354561i\)
\(L(\frac12)\) \(\approx\) \(0.00905092 + 0.354561i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.686 + 0.727i)T \)
3 \( 1 + (1.68 + 0.388i)T \)
good5 \( 1 + (3.71 + 0.434i)T + (4.86 + 1.15i)T^{2} \)
7 \( 1 + (3.25 - 1.63i)T + (4.18 - 5.61i)T^{2} \)
11 \( 1 + (-2.08 + 4.83i)T + (-7.54 - 8.00i)T^{2} \)
13 \( 1 + (0.202 - 0.0480i)T + (11.6 - 5.83i)T^{2} \)
17 \( 1 + (0.290 + 1.64i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-0.286 + 1.62i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (7.81 + 3.92i)T + (13.7 + 18.4i)T^{2} \)
29 \( 1 + (-0.216 - 0.721i)T + (-24.2 + 15.9i)T^{2} \)
31 \( 1 + (4.77 - 3.13i)T + (12.2 - 28.4i)T^{2} \)
37 \( 1 + (-5.15 - 1.87i)T + (28.3 + 23.7i)T^{2} \)
41 \( 1 + (-3.43 - 3.63i)T + (-2.38 + 40.9i)T^{2} \)
43 \( 1 + (0.625 + 0.839i)T + (-12.3 + 41.1i)T^{2} \)
47 \( 1 + (-4.54 - 2.98i)T + (18.6 + 43.1i)T^{2} \)
53 \( 1 + (2.43 + 4.21i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.73 + 10.9i)T + (-40.4 + 42.9i)T^{2} \)
61 \( 1 + (-0.123 + 2.12i)T + (-60.5 - 7.08i)T^{2} \)
67 \( 1 + (-1.79 + 5.98i)T + (-55.9 - 36.8i)T^{2} \)
71 \( 1 + (1.97 - 1.65i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (4.28 + 3.59i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-2.96 + 3.14i)T + (-4.59 - 78.8i)T^{2} \)
83 \( 1 + (9.81 - 10.3i)T + (-4.82 - 82.8i)T^{2} \)
89 \( 1 + (-0.396 - 0.333i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.84 + 0.216i)T + (94.3 - 22.3i)T^{2} \)
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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

−12.24820829997505181692179551791, −11.55229263213760161594430072112, −10.86222048954233765337992825302, −9.448207212497677717045141971783, −8.179189970638095400606179187636, −6.73699818138902261929748084783, −5.84036426994346679103653530145, −4.35250024462410606124283185948, −3.25847412625980890352784494744, −0.31988272978699971607451532457, 3.86575156234538976710887605616, 4.21347112854335364875159638807, 5.99238579768982750031702472148, 7.08898737502398254451601861293, 7.62166467775192985935214291244, 9.475514033873683326817741110817, 10.48432740601327069257462274974, 11.74842718345762088891536175900, 12.24923458307211743096079785931, 13.07768205963622296244105715090

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