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Properties

Label 2-184-8.3-c2-0-4
Degree $2$
Conductor $184$
Sign $-0.853 + 0.521i$
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  + (0.363 + 1.96i)2-s − 0.918·3-s + (−3.73 + 1.43i)4-s + 9.40i·5-s + (−0.333 − 1.80i)6-s − 5.49i·7-s + (−4.17 − 6.82i)8-s − 8.15·9-s + (−18.4 + 3.41i)10-s + 5.58·11-s + (3.42 − 1.31i)12-s + 3.89i·13-s + (10.8 − 1.99i)14-s − 8.63i·15-s + (11.9 − 10.6i)16-s − 8.73·17-s + ⋯
L(s)  = 1  + (0.181 + 0.983i)2-s − 0.306·3-s + (−0.933 + 0.357i)4-s + 1.88i·5-s + (−0.0556 − 0.300i)6-s − 0.784i·7-s + (−0.521 − 0.853i)8-s − 0.906·9-s + (−1.84 + 0.341i)10-s + 0.507·11-s + (0.285 − 0.109i)12-s + 0.299i·13-s + (0.771 − 0.142i)14-s − 0.575i·15-s + (0.744 − 0.667i)16-s − 0.513·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $-0.853 + 0.521i$
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: \((2,\ 184,\ (\ :1),\ -0.853 + 0.521i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.208393 - 0.740850i\)
\(L(\frac12)\) \(\approx\) \(0.208393 - 0.740850i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.363 - 1.96i)T \)
23 \( 1 - 4.79iT \)
good3 \( 1 + 0.918T + 9T^{2} \)
5 \( 1 - 9.40iT - 25T^{2} \)
7 \( 1 + 5.49iT - 49T^{2} \)
11 \( 1 - 5.58T + 121T^{2} \)
13 \( 1 - 3.89iT - 169T^{2} \)
17 \( 1 + 8.73T + 289T^{2} \)
19 \( 1 + 16.2T + 361T^{2} \)
29 \( 1 + 15.1iT - 841T^{2} \)
31 \( 1 - 46.2iT - 961T^{2} \)
37 \( 1 - 30.9iT - 1.36e3T^{2} \)
41 \( 1 - 29.5T + 1.68e3T^{2} \)
43 \( 1 + 23.4T + 1.84e3T^{2} \)
47 \( 1 - 80.0iT - 2.20e3T^{2} \)
53 \( 1 - 9.78iT - 2.80e3T^{2} \)
59 \( 1 - 58.5T + 3.48e3T^{2} \)
61 \( 1 - 112. iT - 3.72e3T^{2} \)
67 \( 1 - 14.4T + 4.48e3T^{2} \)
71 \( 1 + 51.6iT - 5.04e3T^{2} \)
73 \( 1 + 110.T + 5.32e3T^{2} \)
79 \( 1 + 101. iT - 6.24e3T^{2} \)
83 \( 1 - 4.00T + 6.88e3T^{2} \)
89 \( 1 + 162.T + 7.92e3T^{2} \)
97 \( 1 - 84.6T + 9.40e3T^{2} \)
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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

−13.38568436354925350906612005309, −11.86313332787234803837776904020, −10.93560158772469239953449205549, −10.10279930956313209321723738995, −8.739672320514229609395153994949, −7.46321648681809169735799832823, −6.67313995992240640325711489900, −6.00716660335838763118583111321, −4.27196693128689777809998401229, −3.03577020293675780323342187083, 0.43769935216732705723832446244, 2.13858783729416301811373322525, 4.06711949498404833269534651656, 5.17414940022209176777265457865, 5.90931049001628503120586426714, 8.413493375403392154366375868488, 8.783868433509069899162334945567, 9.704575177727391963085322297788, 11.17399825945922454018389848136, 11.90671532064777627812165270665

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