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Properties

Label 2-164-4.3-c2-0-6
Degree $2$
Conductor $164$
Sign $0.924 - 0.380i$
Analytic cond. $4.46867$
Root an. cond. $2.11392$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.387 − 1.96i)2-s − 0.338i·3-s + (−3.69 + 1.52i)4-s − 3.21·5-s + (−0.663 + 0.131i)6-s + 12.6i·7-s + (4.41 + 6.66i)8-s + 8.88·9-s + (1.24 + 6.30i)10-s − 2.51i·11-s + (0.514 + 1.25i)12-s + 6.94·13-s + (24.7 − 4.89i)14-s + 1.08i·15-s + (11.3 − 11.2i)16-s − 13.6·17-s + ⋯
L(s)  = 1  + (−0.193 − 0.981i)2-s − 0.112i·3-s + (−0.924 + 0.380i)4-s − 0.642·5-s + (−0.110 + 0.0218i)6-s + 1.80i·7-s + (0.552 + 0.833i)8-s + 0.987·9-s + (0.124 + 0.630i)10-s − 0.229i·11-s + (0.0428 + 0.104i)12-s + 0.534·13-s + (1.77 − 0.349i)14-s + 0.0724i·15-s + (0.710 − 0.703i)16-s − 0.802·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.924 - 0.380i$
Analytic conductor: \(4.46867\)
Root analytic conductor: \(2.11392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1),\ 0.924 - 0.380i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.977025 + 0.193046i\)
\(L(\frac12)\) \(\approx\) \(0.977025 + 0.193046i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.387 + 1.96i)T \)
41 \( 1 + 6.40T \)
good3 \( 1 + 0.338iT - 9T^{2} \)
5 \( 1 + 3.21T + 25T^{2} \)
7 \( 1 - 12.6iT - 49T^{2} \)
11 \( 1 + 2.51iT - 121T^{2} \)
13 \( 1 - 6.94T + 169T^{2} \)
17 \( 1 + 13.6T + 289T^{2} \)
19 \( 1 - 27.4iT - 361T^{2} \)
23 \( 1 - 9.25iT - 529T^{2} \)
29 \( 1 - 27.8T + 841T^{2} \)
31 \( 1 - 51.3iT - 961T^{2} \)
37 \( 1 - 47.9T + 1.36e3T^{2} \)
43 \( 1 + 7.63iT - 1.84e3T^{2} \)
47 \( 1 - 23.1iT - 2.20e3T^{2} \)
53 \( 1 + 55.7T + 2.80e3T^{2} \)
59 \( 1 + 12.8iT - 3.48e3T^{2} \)
61 \( 1 - 19.2T + 3.72e3T^{2} \)
67 \( 1 + 47.8iT - 4.48e3T^{2} \)
71 \( 1 + 112. iT - 5.04e3T^{2} \)
73 \( 1 + 73.1T + 5.32e3T^{2} \)
79 \( 1 + 69.2iT - 6.24e3T^{2} \)
83 \( 1 - 7.27iT - 6.88e3T^{2} \)
89 \( 1 - 128.T + 7.92e3T^{2} \)
97 \( 1 - 2.67T + 9.40e3T^{2} \)
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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

−12.32388318746269863537362940942, −11.89157068474353194681711263414, −10.80355381265926930614155064610, −9.645262876222118381939336470164, −8.692268087499964685876335084283, −7.88642683817318609496064925992, −6.10125293509167790629224461149, −4.67932072481632217300076027110, −3.31674846669966868163294235106, −1.78444004320737034691437072466, 0.71741140599194399330328833749, 4.07453758771579025786736692944, 4.52532222664603485230580173840, 6.54285366242172280341710812645, 7.26051763880223592859605307830, 8.082809142322720106118342774330, 9.470211061073917707538108113033, 10.37386460222459562166519169739, 11.30805115988824286923603439252, 13.17251155634141157140747351037

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