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Properties

Label 2-164-164.55-c1-0-12
Degree $2$
Conductor $164$
Sign $0.731 - 0.681i$
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  + (1 + i)2-s + 2i·4-s + (2.82 − 2.82i)5-s + (−2 + 2i)8-s + (−2.12 + 2.12i)9-s + 5.65·10-s + (−1.53 − 3.70i)13-s − 4·16-s + (−2.53 + 6.12i)17-s − 4.24·18-s + (5.65 + 5.65i)20-s − 11.0i·25-s + (2.17 − 5.24i)26-s + (−2.87 − 6.94i)29-s + (−4 − 4i)32-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + (1.26 − 1.26i)5-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + 1.78·10-s + (−0.425 − 1.02i)13-s − 16-s + (−0.614 + 1.48i)17-s − 0.999·18-s + (1.26 + 1.26i)20-s − 2.20i·25-s + (0.425 − 1.02i)26-s + (−0.534 − 1.29i)29-s + (−0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.731 - 0.681i$
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: \((2,\ 164,\ (\ :1/2),\ 0.731 - 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60974 + 0.633512i\)
\(L(\frac12)\) \(\approx\) \(1.60974 + 0.633512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
41 \( 1 + (-5 + 4i)T \)
good3 \( 1 + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-2.82 + 2.82i)T - 5iT^{2} \)
7 \( 1 + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.53 + 3.70i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + (2.53 - 6.12i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (13.4 + 13.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (2.87 + 6.94i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7.07T + 37T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (10.5 - 4.36i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-11 - 11i)T + 61iT^{2} \)
67 \( 1 + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \)
79 \( 1 + (-55.8 + 55.8i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-1.19 - 2.87i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + (18.1 + 7.53i)T + (68.5 + 68.5i)T^{2} \)
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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

−13.01723450547124060443414379206, −12.56178101946121257695546333963, −11.09189974231356774452877069006, −9.744561031290349001080475187714, −8.609213461185348737080223403374, −7.900979453486719606298948014531, −6.07602820880771777536947915078, −5.53386972636673754791313266946, −4.41854613690041851993471478884, −2.37028717673295042336630638467, 2.21472590881886127391429385652, 3.24612299227146717500729095449, 5.03730368203500632021883259863, 6.26774163500228694956964770530, 6.91216263780402732744225262033, 9.311147175689424064725476227813, 9.672540383574840590450545124705, 11.03915424755331820524601378566, 11.49062565369214489910011589877, 12.81323584574847026800432482144

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