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Properties

Label 2-184-8.3-c2-0-30
Degree $2$
Conductor $184$
Sign $0.542 + 0.839i$
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  + (1.31 + 1.51i)2-s − 2.44·3-s + (−0.562 + 3.96i)4-s − 7.30i·5-s + (−3.20 − 3.69i)6-s − 8.14i·7-s + (−6.71 + 4.34i)8-s − 3.01·9-s + (11.0 − 9.58i)10-s + 8.07·11-s + (1.37 − 9.68i)12-s − 17.1i·13-s + (12.2 − 10.6i)14-s + 17.8i·15-s + (−15.3 − 4.45i)16-s − 2.64·17-s + ⋯
L(s)  = 1  + (0.655 + 0.755i)2-s − 0.815·3-s + (−0.140 + 0.990i)4-s − 1.46i·5-s + (−0.534 − 0.615i)6-s − 1.16i·7-s + (−0.839 + 0.542i)8-s − 0.334·9-s + (1.10 − 0.958i)10-s + 0.734·11-s + (0.114 − 0.807i)12-s − 1.31i·13-s + (0.878 − 0.762i)14-s + 1.19i·15-s + (−0.960 − 0.278i)16-s − 0.155·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $0.542 + 0.839i$
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: \((2,\ 184,\ (\ :1),\ 0.542 + 0.839i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.07496 - 0.585235i\)
\(L(\frac12)\) \(\approx\) \(1.07496 - 0.585235i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 - 1.51i)T \)
23 \( 1 - 4.79iT \)
good3 \( 1 + 2.44T + 9T^{2} \)
5 \( 1 + 7.30iT - 25T^{2} \)
7 \( 1 + 8.14iT - 49T^{2} \)
11 \( 1 - 8.07T + 121T^{2} \)
13 \( 1 + 17.1iT - 169T^{2} \)
17 \( 1 + 2.64T + 289T^{2} \)
19 \( 1 - 4.64T + 361T^{2} \)
29 \( 1 + 29.7iT - 841T^{2} \)
31 \( 1 - 42.5iT - 961T^{2} \)
37 \( 1 + 41.0iT - 1.36e3T^{2} \)
41 \( 1 + 64.0T + 1.68e3T^{2} \)
43 \( 1 + 27.8T + 1.84e3T^{2} \)
47 \( 1 + 38.4iT - 2.20e3T^{2} \)
53 \( 1 - 36.6iT - 2.80e3T^{2} \)
59 \( 1 - 112.T + 3.48e3T^{2} \)
61 \( 1 - 61.5iT - 3.72e3T^{2} \)
67 \( 1 + 35.4T + 4.48e3T^{2} \)
71 \( 1 + 35.2iT - 5.04e3T^{2} \)
73 \( 1 - 75.4T + 5.32e3T^{2} \)
79 \( 1 - 101. iT - 6.24e3T^{2} \)
83 \( 1 - 100.T + 6.88e3T^{2} \)
89 \( 1 - 165.T + 7.92e3T^{2} \)
97 \( 1 - 44.5T + 9.40e3T^{2} \)
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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

−12.34614048977125694758196724010, −11.63559417272702809449096647022, −10.35247529315803414120836423656, −8.905696570133198565238862414092, −8.033083223409173854151954776824, −6.83100159600670530480748465216, −5.60785259256925488298353269512, −4.88934949437865008104196969065, −3.70439181774039431818638309103, −0.65047018751464242569438285589, 2.16587984628910803365064283956, 3.42106349481219461058866205217, 5.00835241787368171641078575165, 6.23962313549335537074822740262, 6.69152265375686965226703600099, 8.839456359783678554116299012995, 9.886708992268505862515124541840, 10.99455562166083924021927900723, 11.66086098796613845535737442288, 12.01404423196389249709438053720

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