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Properties

Label 2-184-8.3-c2-0-9
Degree $2$
Conductor $184$
Sign $-0.473 - 0.880i$
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  + (−1.97 + 0.327i)2-s + 2.30·3-s + (3.78 − 1.29i)4-s + 6.63i·5-s + (−4.55 + 0.756i)6-s + 5.68i·7-s + (−7.04 + 3.79i)8-s − 3.67·9-s + (−2.17 − 13.0i)10-s − 6.58·11-s + (8.73 − 2.98i)12-s + 1.70i·13-s + (−1.86 − 11.2i)14-s + 15.3i·15-s + (12.6 − 9.78i)16-s − 15.6·17-s + ⋯
L(s)  = 1  + (−0.986 + 0.163i)2-s + 0.769·3-s + (0.946 − 0.323i)4-s + 1.32i·5-s + (−0.759 + 0.126i)6-s + 0.812i·7-s + (−0.880 + 0.473i)8-s − 0.407·9-s + (−0.217 − 1.30i)10-s − 0.598·11-s + (0.728 − 0.248i)12-s + 0.130i·13-s + (−0.133 − 0.801i)14-s + 1.02i·15-s + (0.791 − 0.611i)16-s − 0.922·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $-0.473 - 0.880i$
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: \((2,\ 184,\ (\ :1),\ -0.473 - 0.880i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.500329 + 0.837436i\)
\(L(\frac12)\) \(\approx\) \(0.500329 + 0.837436i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.97 - 0.327i)T \)
23 \( 1 + 4.79iT \)
good3 \( 1 - 2.30T + 9T^{2} \)
5 \( 1 - 6.63iT - 25T^{2} \)
7 \( 1 - 5.68iT - 49T^{2} \)
11 \( 1 + 6.58T + 121T^{2} \)
13 \( 1 - 1.70iT - 169T^{2} \)
17 \( 1 + 15.6T + 289T^{2} \)
19 \( 1 - 17.3T + 361T^{2} \)
29 \( 1 - 28.8iT - 841T^{2} \)
31 \( 1 - 42.9iT - 961T^{2} \)
37 \( 1 + 21.2iT - 1.36e3T^{2} \)
41 \( 1 - 67.2T + 1.68e3T^{2} \)
43 \( 1 - 45.7T + 1.84e3T^{2} \)
47 \( 1 - 81.7iT - 2.20e3T^{2} \)
53 \( 1 + 66.7iT - 2.80e3T^{2} \)
59 \( 1 - 34.9T + 3.48e3T^{2} \)
61 \( 1 + 61.7iT - 3.72e3T^{2} \)
67 \( 1 - 8.27T + 4.48e3T^{2} \)
71 \( 1 - 53.7iT - 5.04e3T^{2} \)
73 \( 1 - 59.7T + 5.32e3T^{2} \)
79 \( 1 + 44.7iT - 6.24e3T^{2} \)
83 \( 1 + 48.5T + 6.88e3T^{2} \)
89 \( 1 - 32.7T + 7.92e3T^{2} \)
97 \( 1 + 132.T + 9.40e3T^{2} \)
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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

−12.55876567770744036800052251539, −11.28080281055456206427157936328, −10.74331758215693856377504827423, −9.499955167145781039448027553398, −8.744399890784124980696863059058, −7.71341799577926818633858855650, −6.77455780284928525168718305116, −5.61460532644938948000880536921, −3.08742214831121804811788616507, −2.36344695028325743021822712470, 0.69533383982255866727368590884, 2.45412255638973998712623397762, 4.08083957937501845952898995081, 5.73170046217235626592049537323, 7.43124858620452939664557119039, 8.143801290697107906178059231347, 9.023559063832599887314467616007, 9.739224495087695054814660148016, 10.95822213314005636169331387358, 11.93359111634804941340963229917

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