Properties

Label 2-162-81.13-c1-0-6
Degree $2$
Conductor $162$
Sign $-0.985 + 0.172i$
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  + (−0.893 − 0.448i)2-s + (−1.41 − 1.00i)3-s + (0.597 + 0.802i)4-s + (0.308 + 1.02i)5-s + (0.814 + 1.52i)6-s + (−2.06 − 4.78i)7-s + (−0.173 − 0.984i)8-s + (0.996 + 2.82i)9-s + (0.186 − 1.05i)10-s + (−5.51 + 1.30i)11-s + (−0.0413 − 1.73i)12-s + (−2.75 − 1.81i)13-s + (−0.303 + 5.20i)14-s + (0.594 − 1.76i)15-s + (−0.286 + 0.957i)16-s + (−1.32 + 0.483i)17-s + ⋯
L(s)  = 1  + (−0.631 − 0.317i)2-s + (−0.816 − 0.577i)3-s + (0.298 + 0.401i)4-s + (0.137 + 0.460i)5-s + (0.332 + 0.624i)6-s + (−0.780 − 1.80i)7-s + (−0.0613 − 0.348i)8-s + (0.332 + 0.943i)9-s + (0.0589 − 0.334i)10-s + (−1.66 + 0.394i)11-s + (−0.0119 − 0.499i)12-s + (−0.765 − 0.503i)13-s + (−0.0810 + 1.39i)14-s + (0.153 − 0.455i)15-s + (−0.0717 + 0.239i)16-s + (−0.322 + 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.985 + 0.172i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ -0.985 + 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0263459 - 0.303633i\)
\(L(\frac12)\) \(\approx\) \(0.0263459 - 0.303633i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.893 + 0.448i)T \)
3 \( 1 + (1.41 + 1.00i)T \)
good5 \( 1 + (-0.308 - 1.02i)T + (-4.17 + 2.74i)T^{2} \)
7 \( 1 + (2.06 + 4.78i)T + (-4.80 + 5.09i)T^{2} \)
11 \( 1 + (5.51 - 1.30i)T + (9.82 - 4.93i)T^{2} \)
13 \( 1 + (2.75 + 1.81i)T + (5.14 + 11.9i)T^{2} \)
17 \( 1 + (1.32 - 0.483i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.526 + 0.191i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-0.185 + 0.429i)T + (-15.7 - 16.7i)T^{2} \)
29 \( 1 + (0.413 + 7.09i)T + (-28.8 + 3.36i)T^{2} \)
31 \( 1 + (-2.69 + 0.314i)T + (30.1 - 7.14i)T^{2} \)
37 \( 1 + (3.28 + 2.75i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (-5.78 + 2.90i)T + (24.4 - 32.8i)T^{2} \)
43 \( 1 + (-6.69 - 7.09i)T + (-2.50 + 42.9i)T^{2} \)
47 \( 1 + (-1.37 - 0.161i)T + (45.7 + 10.8i)T^{2} \)
53 \( 1 + (3.96 - 6.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.06 + 0.252i)T + (52.7 + 26.4i)T^{2} \)
61 \( 1 + (-5.03 + 6.75i)T + (-17.4 - 58.4i)T^{2} \)
67 \( 1 + (0.100 - 1.73i)T + (-66.5 - 7.77i)T^{2} \)
71 \( 1 + (-1.32 + 7.51i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (1.86 + 10.5i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.99 + 1.00i)T + (47.1 + 63.3i)T^{2} \)
83 \( 1 + (10.3 + 5.19i)T + (49.5 + 66.5i)T^{2} \)
89 \( 1 + (0.973 + 5.52i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.471 - 1.57i)T + (-81.0 - 53.3i)T^{2} \)
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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

−12.54591764315023488682955971645, −10.99323342581375986068066223051, −10.47014406506065171973420868364, −9.873064459931335105941212770475, −7.76426259301668671046459346924, −7.34988545036762486303079494445, −6.24009083529781515923218063099, −4.55214756944611784107301805090, −2.65001148016529527745814628992, −0.36359044079345667058916013083, 2.70153864666891934332298071249, 5.10913381130699535969531624078, 5.64414699627172635435127476983, 6.88378759398285987122647463733, 8.540291064299429644134702000723, 9.268204344142566662992600545311, 10.15593662633029151277558754940, 11.23607357362685833259232211919, 12.31394383908970843953410656110, 12.93061333751946483724987958034

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