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Properties

Label 2-184-8.3-c2-0-36
Degree $2$
Conductor $184$
Sign $-0.880 + 0.473i$
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  + (−1.54 + 1.26i)2-s + 0.245·3-s + (0.774 − 3.92i)4-s − 4.88i·5-s + (−0.380 + 0.312i)6-s − 0.635i·7-s + (3.78 + 7.04i)8-s − 8.93·9-s + (6.20 + 7.54i)10-s − 16.7·11-s + (0.190 − 0.965i)12-s + 18.0i·13-s + (0.807 + 0.982i)14-s − 1.20i·15-s + (−14.7 − 6.08i)16-s − 0.253·17-s + ⋯
L(s)  = 1  + (−0.772 + 0.634i)2-s + 0.0819·3-s + (0.193 − 0.981i)4-s − 0.977i·5-s + (−0.0633 + 0.0520i)6-s − 0.0908i·7-s + (0.473 + 0.880i)8-s − 0.993·9-s + (0.620 + 0.754i)10-s − 1.52·11-s + (0.0158 − 0.0804i)12-s + 1.38i·13-s + (0.0576 + 0.0701i)14-s − 0.0801i·15-s + (−0.924 − 0.380i)16-s − 0.0149·17-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0320346 - 0.127312i\)
\(L(\frac12)\) \(\approx\) \(0.0320346 - 0.127312i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.54 - 1.26i)T \)
23 \( 1 + 4.79iT \)
good3 \( 1 - 0.245T + 9T^{2} \)
5 \( 1 + 4.88iT - 25T^{2} \)
7 \( 1 + 0.635iT - 49T^{2} \)
11 \( 1 + 16.7T + 121T^{2} \)
13 \( 1 - 18.0iT - 169T^{2} \)
17 \( 1 + 0.253T + 289T^{2} \)
19 \( 1 + 24.4T + 361T^{2} \)
29 \( 1 + 52.8iT - 841T^{2} \)
31 \( 1 - 29.9iT - 961T^{2} \)
37 \( 1 + 28.6iT - 1.36e3T^{2} \)
41 \( 1 + 51.8T + 1.68e3T^{2} \)
43 \( 1 + 63.6T + 1.84e3T^{2} \)
47 \( 1 - 39.3iT - 2.20e3T^{2} \)
53 \( 1 + 58.7iT - 2.80e3T^{2} \)
59 \( 1 - 84.9T + 3.48e3T^{2} \)
61 \( 1 - 14.4iT - 3.72e3T^{2} \)
67 \( 1 + 2.53T + 4.48e3T^{2} \)
71 \( 1 + 65.9iT - 5.04e3T^{2} \)
73 \( 1 - 38.9T + 5.32e3T^{2} \)
79 \( 1 - 62.9iT - 6.24e3T^{2} \)
83 \( 1 + 129.T + 6.88e3T^{2} \)
89 \( 1 + 103.T + 7.92e3T^{2} \)
97 \( 1 - 118.T + 9.40e3T^{2} \)
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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

−11.81480073522734965990778866794, −10.81677100973010487804406288076, −9.764368080432886094972375579982, −8.606953970261408700754960606924, −8.268418551048781733013315126159, −6.83019333549438021392963703373, −5.62224440478282653214114335843, −4.61839675602351183139874355789, −2.18547608345808087049203561851, −0.089925096311959133658249209108, 2.51597744067916206406780932360, 3.27704716673860442831375341149, 5.36666415115206113963517166632, 6.83991439442365208726352304078, 7.998779438617783722355656685040, 8.648275522856806099003511902503, 10.28981352776055365518232369068, 10.55416752596380204991448082653, 11.51525841612984607356125628175, 12.73216552364958555094841678381

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