Properties

Label 2-162-81.5-c2-0-13
Degree $2$
Conductor $162$
Sign $0.685 + 0.728i$
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  + (0.326 + 1.37i)2-s + (−0.0281 − 2.99i)3-s + (−1.78 + 0.897i)4-s + (1.73 − 1.28i)5-s + (4.11 − 1.01i)6-s + (3.67 − 2.41i)7-s + (−1.81 − 2.16i)8-s + (−8.99 + 0.168i)9-s + (2.33 + 1.96i)10-s + (1.53 − 13.1i)11-s + (2.74 + 5.33i)12-s + (−2.24 − 7.51i)13-s + (4.52 + 4.27i)14-s + (−3.91 − 5.15i)15-s + (2.38 − 3.20i)16-s + (32.0 + 5.64i)17-s + ⋯
L(s)  = 1  + (0.163 + 0.688i)2-s + (−0.00937 − 0.999i)3-s + (−0.446 + 0.224i)4-s + (0.346 − 0.257i)5-s + (0.686 − 0.169i)6-s + (0.525 − 0.345i)7-s + (−0.227 − 0.270i)8-s + (−0.999 + 0.0187i)9-s + (0.233 + 0.196i)10-s + (0.139 − 1.19i)11-s + (0.228 + 0.444i)12-s + (−0.173 − 0.577i)13-s + (0.323 + 0.305i)14-s + (−0.261 − 0.343i)15-s + (0.149 − 0.200i)16-s + (1.88 + 0.332i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.685 + 0.728i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ 0.685 + 0.728i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.44392 - 0.623954i\)
\(L(\frac12)\) \(\approx\) \(1.44392 - 0.623954i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.326 - 1.37i)T \)
3 \( 1 + (0.0281 + 2.99i)T \)
good5 \( 1 + (-1.73 + 1.28i)T + (7.17 - 23.9i)T^{2} \)
7 \( 1 + (-3.67 + 2.41i)T + (19.4 - 44.9i)T^{2} \)
11 \( 1 + (-1.53 + 13.1i)T + (-117. - 27.9i)T^{2} \)
13 \( 1 + (2.24 + 7.51i)T + (-141. + 92.8i)T^{2} \)
17 \( 1 + (-32.0 - 5.64i)T + (271. + 98.8i)T^{2} \)
19 \( 1 + (4.44 + 25.2i)T + (-339. + 123. i)T^{2} \)
23 \( 1 + (2.91 - 4.43i)T + (-209. - 485. i)T^{2} \)
29 \( 1 + (-7.70 + 7.26i)T + (48.8 - 839. i)T^{2} \)
31 \( 1 + (1.26 - 21.7i)T + (-954. - 111. i)T^{2} \)
37 \( 1 + (43.5 - 15.8i)T + (1.04e3 - 879. i)T^{2} \)
41 \( 1 + (3.35 - 14.1i)T + (-1.50e3 - 754. i)T^{2} \)
43 \( 1 + (-5.82 - 13.4i)T + (-1.26e3 + 1.34e3i)T^{2} \)
47 \( 1 + (-91.8 + 5.34i)T + (2.19e3 - 256. i)T^{2} \)
53 \( 1 + (29.2 + 16.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-3.98 - 34.0i)T + (-3.38e3 + 802. i)T^{2} \)
61 \( 1 + (25.1 + 12.6i)T + (2.22e3 + 2.98e3i)T^{2} \)
67 \( 1 + (62.5 - 66.2i)T + (-261. - 4.48e3i)T^{2} \)
71 \( 1 + (-21.0 + 25.1i)T + (-875. - 4.96e3i)T^{2} \)
73 \( 1 + (39.6 - 33.2i)T + (925. - 5.24e3i)T^{2} \)
79 \( 1 + (-117. + 27.8i)T + (5.57e3 - 2.80e3i)T^{2} \)
83 \( 1 + (-2.39 - 10.0i)T + (-6.15e3 + 3.09e3i)T^{2} \)
89 \( 1 + (-85.7 - 102. i)T + (-1.37e3 + 7.80e3i)T^{2} \)
97 \( 1 + (-56.0 + 75.3i)T + (-2.69e3 - 9.01e3i)T^{2} \)
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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

−12.71251670459470922200791821603, −11.74311575166041454305555112643, −10.58307013788132067428815813662, −9.045218202302183467264545893434, −8.114189536651798780678508403924, −7.30229225547843938795342254827, −6.02130752962667915115408995224, −5.20860187205088687390825630570, −3.21836044366142509943133480177, −1.03918307165922010819462590687, 2.09833549274603264136529858661, 3.69065471905353465999147877666, 4.85740394685129916583530799584, 5.91122865989110473321558757676, 7.75076336938047870586536594791, 9.060987486572342076590346358035, 10.00325118939323622732420487215, 10.49629079421830015744427998913, 11.94192402723243276367188658404, 12.26240212333551050659380409087

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