Properties

Label 2-164-164.111-c1-0-16
Degree $2$
Conductor $164$
Sign $0.722 + 0.691i$
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  + (0.852 − 1.12i)2-s + (2.24 + 0.930i)3-s + (−0.545 − 1.92i)4-s + (−1.19 + 0.607i)5-s + (2.96 − 1.74i)6-s + (−0.861 − 1.40i)7-s + (−2.63 − 1.02i)8-s + (2.06 + 2.06i)9-s + (−0.331 + 1.86i)10-s + (1.90 + 0.150i)11-s + (0.566 − 4.83i)12-s + (0.816 + 3.40i)13-s + (−2.32 − 0.227i)14-s + (−3.24 + 0.255i)15-s + (−3.40 + 2.09i)16-s + (−4.46 + 3.81i)17-s + ⋯
L(s)  = 1  + (0.603 − 0.797i)2-s + (1.29 + 0.537i)3-s + (−0.272 − 0.962i)4-s + (−0.533 + 0.271i)5-s + (1.21 − 0.710i)6-s + (−0.325 − 0.531i)7-s + (−0.931 − 0.362i)8-s + (0.686 + 0.686i)9-s + (−0.104 + 0.589i)10-s + (0.575 + 0.0452i)11-s + (0.163 − 1.39i)12-s + (0.226 + 0.943i)13-s + (−0.620 − 0.0607i)14-s + (−0.838 + 0.0659i)15-s + (−0.851 + 0.524i)16-s + (−1.08 + 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74035 - 0.698997i\)
\(L(\frac12)\) \(\approx\) \(1.74035 - 0.698997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.852 + 1.12i)T \)
41 \( 1 + (-6.30 - 1.14i)T \)
good3 \( 1 + (-2.24 - 0.930i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (1.19 - 0.607i)T + (2.93 - 4.04i)T^{2} \)
7 \( 1 + (0.861 + 1.40i)T + (-3.17 + 6.23i)T^{2} \)
11 \( 1 + (-1.90 - 0.150i)T + (10.8 + 1.72i)T^{2} \)
13 \( 1 + (-0.816 - 3.40i)T + (-11.5 + 5.90i)T^{2} \)
17 \( 1 + (4.46 - 3.81i)T + (2.65 - 16.7i)T^{2} \)
19 \( 1 + (-1.57 - 0.378i)T + (16.9 + 8.62i)T^{2} \)
23 \( 1 + (-1.05 + 0.763i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (7.58 + 6.47i)T + (4.53 + 28.6i)T^{2} \)
31 \( 1 + (2.13 + 6.57i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.802 + 2.46i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-10.9 - 1.72i)T + (40.8 + 13.2i)T^{2} \)
47 \( 1 + (2.02 - 3.30i)T + (-21.3 - 41.8i)T^{2} \)
53 \( 1 + (2.84 - 3.32i)T + (-8.29 - 52.3i)T^{2} \)
59 \( 1 + (-3.96 - 5.45i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-8.73 + 1.38i)T + (58.0 - 18.8i)T^{2} \)
67 \( 1 + (1.15 + 14.6i)T + (-66.1 + 10.4i)T^{2} \)
71 \( 1 + (0.561 - 7.13i)T + (-70.1 - 11.1i)T^{2} \)
73 \( 1 + (-1.44 + 1.44i)T - 73iT^{2} \)
79 \( 1 + (-2.46 + 5.93i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + 0.232iT - 83T^{2} \)
89 \( 1 + (-3.81 + 2.33i)T + (40.4 - 79.2i)T^{2} \)
97 \( 1 + (-0.857 - 10.8i)T + (-95.8 + 15.1i)T^{2} \)
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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

−12.99367983239386486097838132168, −11.58202374603846228683269764852, −10.86008237321861111400635172556, −9.543337759462737988439050319304, −9.093301062930142275076547465910, −7.63403821236547707480294967366, −6.20460482697797822014341778765, −4.05246677249651871571534945757, −3.88611392256108615993013577267, −2.25148433093616978995672423353, 2.72458277737927807484164677658, 3.82591469496825160126567573295, 5.39581626530268934769307583834, 6.87620753586492144768341534020, 7.70353030745270479750194072480, 8.719310476759971104164820616602, 9.233981581487372079599696331034, 11.31713961994780278372236422702, 12.50544948258733515514851264119, 13.06578046384277784809971561900

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