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Properties

Label 2-164-164.55-c1-0-8
Degree $2$
Conductor $164$
Sign $-0.206 + 0.978i$
Analytic cond. $1.30954$
Root an. cond. $1.14435$
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  + (−0.819 − 1.15i)2-s + (−1.81 + 0.752i)3-s + (−0.658 + 1.88i)4-s + (1.86 − 1.86i)5-s + (2.35 + 1.47i)6-s + (0.905 − 0.375i)7-s + (2.71 − 0.788i)8-s + (0.614 − 0.614i)9-s + (−3.67 − 0.621i)10-s + (−2.28 − 5.51i)11-s + (−0.225 − 3.92i)12-s + (−1.33 − 3.22i)13-s + (−1.17 − 0.736i)14-s + (−1.98 + 4.78i)15-s + (−3.13 − 2.48i)16-s + (1.64 − 3.95i)17-s + ⋯
L(s)  = 1  + (−0.579 − 0.815i)2-s + (−1.04 + 0.434i)3-s + (−0.329 + 0.944i)4-s + (0.833 − 0.833i)5-s + (0.961 + 0.603i)6-s + (0.342 − 0.141i)7-s + (0.960 − 0.278i)8-s + (0.204 − 0.204i)9-s + (−1.16 − 0.196i)10-s + (−0.688 − 1.66i)11-s + (−0.0650 − 1.13i)12-s + (−0.370 − 0.895i)13-s + (−0.313 − 0.196i)14-s + (−0.512 + 1.23i)15-s + (−0.783 − 0.621i)16-s + (0.397 − 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.206 + 0.978i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ -0.206 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.400481 - 0.493694i\)
\(L(\frac12)\) \(\approx\) \(0.400481 - 0.493694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.819 + 1.15i)T \)
41 \( 1 + (-0.335 - 6.39i)T \)
good3 \( 1 + (1.81 - 0.752i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.86 + 1.86i)T - 5iT^{2} \)
7 \( 1 + (-0.905 + 0.375i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (2.28 + 5.51i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.33 + 3.22i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + (-1.64 + 3.95i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-5.77 - 2.39i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 - 3.49T + 23T^{2} \)
29 \( 1 + (-1.43 - 3.47i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 8.08T + 31T^{2} \)
37 \( 1 + 4.05T + 37T^{2} \)
43 \( 1 + (-4.80 + 4.80i)T - 43iT^{2} \)
47 \( 1 + (-1.17 - 0.486i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (-4.34 + 1.80i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 - 0.513iT - 59T^{2} \)
61 \( 1 + (-1.39 - 1.39i)T + 61iT^{2} \)
67 \( 1 + (0.603 + 0.249i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (3.07 - 1.27i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.59 + 2.59i)T + 73iT^{2} \)
79 \( 1 + (-3.72 - 8.99i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 - 3.86iT - 83T^{2} \)
89 \( 1 + (-0.771 - 1.86i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + (-14.2 - 5.91i)T + (68.5 + 68.5i)T^{2} \)
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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

−12.32833107400441982851531993962, −11.32138793762785398018878197747, −10.67481236585027516893776629329, −9.745770199685197938889726661936, −8.736339677525672084107606342070, −7.61446221832501251413181182367, −5.52527546335614692093210544431, −5.14019383868231151835242830804, −3.11739517427002935944497510761, −0.874593781547764608349301623232, 1.92078796093324024442343574250, 4.92294419061843059612444988328, 5.81115651939244614209850213904, 6.89864389752376165133143908698, 7.44401115320647382458569480722, 9.191794073144860435225617304208, 10.09576167562866465728160138548, 10.91028888689821061244740148159, 12.03727465832909526564480560073, 13.17449966609752191576061275859

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