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Properties

Label 1-184-184.91-r0-0-0
Degree $1$
Conductor $184$
Sign $1$
Analytic cond. $0.854492$
Root an. cond. $0.854492$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 7-s + 9-s − 11-s − 13-s + 15-s − 17-s − 19-s + 21-s + 25-s + 27-s − 29-s − 31-s − 33-s + 35-s + 37-s − 39-s + 41-s − 43-s + 45-s − 47-s + 49-s − 51-s + 53-s − 55-s − 57-s + ⋯
L(s)  = 1  + 3-s + 5-s + 7-s + 9-s − 11-s − 13-s + 15-s − 17-s − 19-s + 21-s + 25-s + 27-s − 29-s − 31-s − 33-s + 35-s + 37-s − 39-s + 41-s − 43-s + 45-s − 47-s + 49-s − 51-s + 53-s − 55-s − 57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $1$
Analytic conductor: \(0.854492\)
Root analytic conductor: \(0.854492\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{184} (91, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.823197885\)
\(L(\frac12)\) \(\approx\) \(1.823197885\)
\(L(1)\) \(\approx\) \(1.591314213\)
\(L(1)\) \(\approx\) \(1.591314213\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.97062676728454625820749944630, −26.26618081386199348954886808273, −25.33904575679661682917128421044, −24.48191684265106356089084962014, −23.80951153403079438680610439848, −22.08460019370580418446850213930, −21.32395078800824931736445496786, −20.64830067613350781638876552411, −19.66272669811001978120479774200, −18.39514151695679459452616393411, −17.75818745479263995016741760193, −16.56069195612167461063234640001, −15.04700725263228969328672147834, −14.60220539839141746565952176787, −13.40866857885728857059081149390, −12.78778663064352138333096177359, −11.05405769753857887538690718329, −10.058404182478627759651902456690, −9.03335340810330335373498655935, −8.04711331193434781834210360590, −6.9819532799260372174711376447, −5.38846429248504426113080326447, −4.34886994907457533484751287298, −2.54549262523856926627354562678, −1.90198059435967281089087412698, 1.90198059435967281089087412698, 2.54549262523856926627354562678, 4.34886994907457533484751287298, 5.38846429248504426113080326447, 6.9819532799260372174711376447, 8.04711331193434781834210360590, 9.03335340810330335373498655935, 10.058404182478627759651902456690, 11.05405769753857887538690718329, 12.78778663064352138333096177359, 13.40866857885728857059081149390, 14.60220539839141746565952176787, 15.04700725263228969328672147834, 16.56069195612167461063234640001, 17.75818745479263995016741760193, 18.39514151695679459452616393411, 19.66272669811001978120479774200, 20.64830067613350781638876552411, 21.32395078800824931736445496786, 22.08460019370580418446850213930, 23.80951153403079438680610439848, 24.48191684265106356089084962014, 25.33904575679661682917128421044, 26.26618081386199348954886808273, 26.97062676728454625820749944630

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