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Properties

Label 2-927-103.102-c0-0-5
Degree $2$
Conductor $927$
Sign $1$
Analytic cond. $0.462633$
Root an. cond. $0.680171$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·2-s + 1.61·4-s + 0.618·7-s + 8-s − 1.61·13-s + 1.00·14-s − 0.618·17-s − 1.61·19-s + 1.61·23-s + 25-s − 2.61·26-s + 1.00·28-s − 0.618·29-s − 32-s − 1.00·34-s − 2.61·38-s + 1.61·41-s + 2.61·46-s − 0.618·49-s + 1.61·50-s − 2.61·52-s + 0.618·56-s − 1.00·58-s − 0.618·59-s + 0.618·61-s − 1.61·64-s − 1.00·68-s + ⋯
L(s)  = 1  + 1.61·2-s + 1.61·4-s + 0.618·7-s + 8-s − 1.61·13-s + 1.00·14-s − 0.618·17-s − 1.61·19-s + 1.61·23-s + 25-s − 2.61·26-s + 1.00·28-s − 0.618·29-s − 32-s − 1.00·34-s − 2.61·38-s + 1.61·41-s + 2.61·46-s − 0.618·49-s + 1.61·50-s − 2.61·52-s + 0.618·56-s − 1.00·58-s − 0.618·59-s + 0.618·61-s − 1.61·64-s − 1.00·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $1$
Analytic conductor: \(0.462633\)
Root analytic conductor: \(0.680171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (514, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.270258447\)
\(L(\frac12)\) \(\approx\) \(2.270258447\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 - T \)
good2 \( 1 - 1.61T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - 0.618T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.61T + T^{2} \)
17 \( 1 + 0.618T + T^{2} \)
19 \( 1 + 1.61T + T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
29 \( 1 + 0.618T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.61T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 0.618T + T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.61T + T^{2} \)
83 \( 1 + 0.618T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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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

−10.77261549186022846070836068324, −9.453046778798123245451729413043, −8.555809977384639433432940183809, −7.32420320383045325708279180575, −6.74838133129326370058408346152, −5.68229934267069263904404091702, −4.74780865900708731254552737730, −4.39280252164025836403199604291, −2.97671953618363196033058754047, −2.13516838096298957613053787995, 2.13516838096298957613053787995, 2.97671953618363196033058754047, 4.39280252164025836403199604291, 4.74780865900708731254552737730, 5.68229934267069263904404091702, 6.74838133129326370058408346152, 7.32420320383045325708279180575, 8.555809977384639433432940183809, 9.453046778798123245451729413043, 10.77261549186022846070836068324

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