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Properties

Label 2-184-8.3-c2-0-22
Degree $2$
Conductor $184$
Sign $0.983 - 0.181i$
Analytic cond. $5.01363$
Root an. cond. $2.23911$
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  + (−1.66 + 1.10i)2-s + 2.74·3-s + (1.56 − 3.68i)4-s − 5.36i·5-s + (−4.58 + 3.03i)6-s + 10.3i·7-s + (1.45 + 7.86i)8-s − 1.44·9-s + (5.92 + 8.94i)10-s + 19.0·11-s + (4.29 − 10.1i)12-s − 18.2i·13-s + (−11.3 − 17.2i)14-s − 14.7i·15-s + (−11.1 − 11.5i)16-s + 18.3·17-s + ⋯
L(s)  = 1  + (−0.833 + 0.551i)2-s + 0.916·3-s + (0.390 − 0.920i)4-s − 1.07i·5-s + (−0.763 + 0.505i)6-s + 1.47i·7-s + (0.181 + 0.983i)8-s − 0.160·9-s + (0.592 + 0.894i)10-s + 1.73·11-s + (0.358 − 0.843i)12-s − 1.40i·13-s + (−0.813 − 1.22i)14-s − 0.982i·15-s + (−0.694 − 0.719i)16-s + 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $0.983 - 0.181i$
Analytic conductor: \(5.01363\)
Root analytic conductor: \(2.23911\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 184,\ (\ :1),\ 0.983 - 0.181i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.44674 + 0.132694i\)
\(L(\frac12)\) \(\approx\) \(1.44674 + 0.132694i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.66 - 1.10i)T \)
23 \( 1 - 4.79iT \)
good3 \( 1 - 2.74T + 9T^{2} \)
5 \( 1 + 5.36iT - 25T^{2} \)
7 \( 1 - 10.3iT - 49T^{2} \)
11 \( 1 - 19.0T + 121T^{2} \)
13 \( 1 + 18.2iT - 169T^{2} \)
17 \( 1 - 18.3T + 289T^{2} \)
19 \( 1 - 23.0T + 361T^{2} \)
29 \( 1 + 1.54iT - 841T^{2} \)
31 \( 1 + 23.3iT - 961T^{2} \)
37 \( 1 - 16.0iT - 1.36e3T^{2} \)
41 \( 1 + 13.8T + 1.68e3T^{2} \)
43 \( 1 + 79.9T + 1.84e3T^{2} \)
47 \( 1 - 72.3iT - 2.20e3T^{2} \)
53 \( 1 - 29.7iT - 2.80e3T^{2} \)
59 \( 1 - 37.7T + 3.48e3T^{2} \)
61 \( 1 + 4.27iT - 3.72e3T^{2} \)
67 \( 1 + 65.1T + 4.48e3T^{2} \)
71 \( 1 + 27.3iT - 5.04e3T^{2} \)
73 \( 1 + 64.7T + 5.32e3T^{2} \)
79 \( 1 + 91.3iT - 6.24e3T^{2} \)
83 \( 1 - 110.T + 6.88e3T^{2} \)
89 \( 1 + 140.T + 7.92e3T^{2} \)
97 \( 1 + 82.7T + 9.40e3T^{2} \)
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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

−12.22029645695723119770023535661, −11.60532686548506010678020220954, −9.773427408398929231100762301476, −9.173260336878660447905207500576, −8.505886108128907729553519911337, −7.75534714282865834739801527513, −6.02315803214314651290539155049, −5.23101820031642315684688810775, −3.07471721754731921186093096742, −1.33087346584167417155959554008, 1.49639231164429394527562810830, 3.26479824202040688931638145599, 3.89444702055710733233744856267, 6.80949447689831244672002112758, 7.16480657241857905728130542246, 8.454557344367867288777543185724, 9.487332924125204774249167902077, 10.17777601328807253847214078825, 11.34248263110411880116720979418, 11.90416615346900384359661963862

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