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Properties

Label 2-184-8.3-c2-0-39
Degree $2$
Conductor $184$
Sign $-0.520 + 0.853i$
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  + (1.29 − 1.52i)2-s + 0.723·3-s + (−0.630 − 3.94i)4-s − 1.34i·5-s + (0.939 − 1.10i)6-s − 5.69i·7-s + (−6.82 − 4.16i)8-s − 8.47·9-s + (−2.05 − 1.75i)10-s + 10.7·11-s + (−0.456 − 2.85i)12-s − 6.75i·13-s + (−8.66 − 7.39i)14-s − 0.976i·15-s + (−15.2 + 4.98i)16-s + 23.0·17-s + ⋯
L(s)  = 1  + (0.648 − 0.760i)2-s + 0.241·3-s + (−0.157 − 0.987i)4-s − 0.269i·5-s + (0.156 − 0.183i)6-s − 0.813i·7-s + (−0.853 − 0.520i)8-s − 0.941·9-s + (−0.205 − 0.175i)10-s + 0.977·11-s + (−0.0380 − 0.238i)12-s − 0.519i·13-s + (−0.618 − 0.527i)14-s − 0.0650i·15-s + (−0.950 + 0.311i)16-s + 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $-0.520 + 0.853i$
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: \((2,\ 184,\ (\ :1),\ -0.520 + 0.853i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.00013 - 1.78190i\)
\(L(\frac12)\) \(\approx\) \(1.00013 - 1.78190i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 + 1.52i)T \)
23 \( 1 - 4.79iT \)
good3 \( 1 - 0.723T + 9T^{2} \)
5 \( 1 + 1.34iT - 25T^{2} \)
7 \( 1 + 5.69iT - 49T^{2} \)
11 \( 1 - 10.7T + 121T^{2} \)
13 \( 1 + 6.75iT - 169T^{2} \)
17 \( 1 - 23.0T + 289T^{2} \)
19 \( 1 + 11.2T + 361T^{2} \)
29 \( 1 + 34.7iT - 841T^{2} \)
31 \( 1 - 36.3iT - 961T^{2} \)
37 \( 1 - 52.8iT - 1.36e3T^{2} \)
41 \( 1 - 41.9T + 1.68e3T^{2} \)
43 \( 1 - 26.7T + 1.84e3T^{2} \)
47 \( 1 - 16.3iT - 2.20e3T^{2} \)
53 \( 1 - 2.07iT - 2.80e3T^{2} \)
59 \( 1 + 22.9T + 3.48e3T^{2} \)
61 \( 1 + 97.3iT - 3.72e3T^{2} \)
67 \( 1 - 19.3T + 4.48e3T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 - 106.T + 5.32e3T^{2} \)
79 \( 1 + 10.2iT - 6.24e3T^{2} \)
83 \( 1 + 162.T + 6.88e3T^{2} \)
89 \( 1 + 50.9T + 7.92e3T^{2} \)
97 \( 1 - 2.03T + 9.40e3T^{2} \)
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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

−12.10315471545946541580094483172, −11.18529894159493936750546485440, −10.24117886541574080666746681352, −9.258466564942252172972789210064, −8.105587603593204337546012102881, −6.55877500213143980130982623612, −5.40551680866807241677299498364, −4.08954211816235260457706772265, −2.97982466421599666194045424493, −1.05291062056573247243732909767, 2.64812407539974842321120857993, 3.91366772182581384473678618142, 5.44944651707755227369359738684, 6.27542663723980314566176343323, 7.46341343997660682852384745556, 8.664730850118713042159678158612, 9.260319630125025434408412642693, 11.05123896768529531202718866421, 12.01136999636088347845164757492, 12.67470631582071025556482311622

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