Properties

Label 2-162-9.5-c2-0-3
Degree $2$
Conductor $162$
Sign $0.996 + 0.0871i$
Analytic cond. $4.41418$
Root an. cond. $2.10099$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (5.01 + 2.89i)5-s + (4.19 + 7.26i)7-s − 2.82i·8-s + 8.19·10-s + (−12.7 + 7.34i)11-s + (10.5 − 18.3i)13-s + (10.2 + 5.93i)14-s + (−2.00 − 3.46i)16-s − 7.76i·17-s + 24.3·19-s + (10.0 − 5.79i)20-s + (−10.3 + 18i)22-s + (−12.7 − 7.34i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (1.00 + 0.579i)5-s + (0.599 + 1.03i)7-s − 0.353i·8-s + 0.819·10-s + (−1.15 + 0.668i)11-s + (0.815 − 1.41i)13-s + (0.734 + 0.423i)14-s + (−0.125 − 0.216i)16-s − 0.456i·17-s + 1.28·19-s + (0.501 − 0.289i)20-s + (−0.472 + 0.818i)22-s + (−0.553 − 0.319i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.996 + 0.0871i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ 0.996 + 0.0871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.38017 - 0.103920i\)
\(L(\frac12)\) \(\approx\) \(2.38017 - 0.103920i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (-5.01 - 2.89i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.19 - 7.26i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (12.7 - 7.34i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-10.5 + 18.3i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 7.76iT - 289T^{2} \)
19 \( 1 - 24.3T + 361T^{2} \)
23 \( 1 + (12.7 + 7.34i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (30.7 - 17.7i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 60.5T + 1.36e3T^{2} \)
41 \( 1 + (29.1 + 16.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (4.58 + 7.94i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-14.6 + 8.48i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 25.7iT - 2.80e3T^{2} \)
59 \( 1 + (-53.4 - 30.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (10.5 - 18.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 101. iT - 5.04e3T^{2} \)
73 \( 1 - 40.4T + 5.32e3T^{2} \)
79 \( 1 + (49.3 + 85.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (89.6 - 51.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 134. iT - 7.92e3T^{2} \)
97 \( 1 + (-37.5 - 65.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.71776490848737214614341546489, −11.70081444650877486193612518794, −10.56783015269917616463180044847, −9.946721853404656125421158301756, −8.552252249062906275791491827427, −7.22528954067848970566625917364, −5.60485542582848396454439465776, −5.32812207025694996889567957564, −3.13177559173440555740712333541, −2.02521181221762547411533141532, 1.68394490542215810343124684666, 3.74682708853466206035149213783, 5.06354840834366759172966862329, 5.97343949287909813169713164627, 7.31396024137372069418018691225, 8.379665574223430770466059092705, 9.591000751536001290437900115041, 10.77285591005157821007219172794, 11.66746401187775735211438801012, 13.07054793764025039839571513447

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