Properties

Label 2-162-81.13-c1-0-0
Degree $2$
Conductor $162$
Sign $0.999 - 0.0323i$
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 + (−1.71 + 0.231i)3-s + (0.597 + 0.802i)4-s + (−0.267 − 0.892i)5-s + (1.63 + 0.563i)6-s + (1.11 + 2.57i)7-s + (−0.173 − 0.984i)8-s + (2.89 − 0.795i)9-s + (−0.161 + 0.917i)10-s + (4.46 − 1.05i)11-s + (−1.21 − 1.23i)12-s + (1.47 + 0.969i)13-s + (0.163 − 2.80i)14-s + (0.665 + 1.46i)15-s + (−0.286 + 0.957i)16-s + (1.42 − 0.518i)17-s + ⋯
L(s)  = 1  + (−0.631 − 0.317i)2-s + (−0.991 + 0.133i)3-s + (0.298 + 0.401i)4-s + (−0.119 − 0.399i)5-s + (0.668 + 0.229i)6-s + (0.420 + 0.974i)7-s + (−0.0613 − 0.348i)8-s + (0.964 − 0.265i)9-s + (−0.0511 + 0.290i)10-s + (1.34 − 0.318i)11-s + (−0.349 − 0.357i)12-s + (0.408 + 0.268i)13-s + (0.0436 − 0.748i)14-s + (0.171 + 0.379i)15-s + (−0.0717 + 0.239i)16-s + (0.345 − 0.125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.999 - 0.0323i$
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: \((2,\ 162,\ (\ :1/2),\ 0.999 - 0.0323i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.706721 + 0.0114210i\)
\(L(\frac12)\) \(\approx\) \(0.706721 + 0.0114210i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 0.231i)T \)
good5 \( 1 + (0.267 + 0.892i)T + (-4.17 + 2.74i)T^{2} \)
7 \( 1 + (-1.11 - 2.57i)T + (-4.80 + 5.09i)T^{2} \)
11 \( 1 + (-4.46 + 1.05i)T + (9.82 - 4.93i)T^{2} \)
13 \( 1 + (-1.47 - 0.969i)T + (5.14 + 11.9i)T^{2} \)
17 \( 1 + (-1.42 + 0.518i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-4.12 - 1.50i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (2.92 - 6.77i)T + (-15.7 - 16.7i)T^{2} \)
29 \( 1 + (0.418 + 7.19i)T + (-28.8 + 3.36i)T^{2} \)
31 \( 1 + (6.24 - 0.730i)T + (30.1 - 7.14i)T^{2} \)
37 \( 1 + (0.516 + 0.433i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (8.46 - 4.25i)T + (24.4 - 32.8i)T^{2} \)
43 \( 1 + (-3.77 - 4.00i)T + (-2.50 + 42.9i)T^{2} \)
47 \( 1 + (-5.01 - 0.586i)T + (45.7 + 10.8i)T^{2} \)
53 \( 1 + (1.33 - 2.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.89 - 1.16i)T + (52.7 + 26.4i)T^{2} \)
61 \( 1 + (-3.14 + 4.22i)T + (-17.4 - 58.4i)T^{2} \)
67 \( 1 + (-0.832 + 14.2i)T + (-66.5 - 7.77i)T^{2} \)
71 \( 1 + (0.0492 - 0.279i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (2.01 + 11.4i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-3.17 - 1.59i)T + (47.1 + 63.3i)T^{2} \)
83 \( 1 + (9.56 + 4.80i)T + (49.5 + 66.5i)T^{2} \)
89 \( 1 + (-1.74 - 9.90i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (4.96 - 16.5i)T + (-81.0 - 53.3i)T^{2} \)
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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

−12.25686188625362207625278362148, −11.82016498187471380960532433780, −11.13142784300284100315018494617, −9.725526827923773251948147659701, −9.040403904284734993531093721516, −7.78762591463296461260114984529, −6.38658359153804242816625942365, −5.36218192528410021012739294950, −3.81838966245732108840173169052, −1.44494644829390451379095448270, 1.22191347438888706362493032815, 3.98864540282419771970954605726, 5.44169159369049888110797398104, 6.84029094606588091425388920863, 7.24114097473376897909417948495, 8.745953283705771341158733380909, 10.07520090629794252456980481950, 10.80128036638914143954388612312, 11.60035557697014952500343775685, 12.64150641391354428955428214647

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