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Properties

Label 2-927-103.102-c0-0-6
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  + 1.90·2-s + 2.61·4-s − 1.61·7-s + 3.07·8-s − 0.618·13-s − 3.07·14-s + 3.23·16-s − 1.17·17-s + 0.618·19-s − 1.90·23-s + 25-s − 1.17·26-s − 4.23·28-s + 1.17·29-s + 3.07·32-s − 2.23·34-s + 1.17·38-s − 1.90·41-s − 3.61·46-s + 1.61·49-s + 1.90·50-s − 1.61·52-s − 4.97·56-s + 2.23·58-s + 1.17·59-s + 1.61·61-s + 2.61·64-s + ⋯
L(s)  = 1  + 1.90·2-s + 2.61·4-s − 1.61·7-s + 3.07·8-s − 0.618·13-s − 3.07·14-s + 3.23·16-s − 1.17·17-s + 0.618·19-s − 1.90·23-s + 25-s − 1.17·26-s − 4.23·28-s + 1.17·29-s + 3.07·32-s − 2.23·34-s + 1.17·38-s − 1.90·41-s − 3.61·46-s + 1.61·49-s + 1.90·50-s − 1.61·52-s − 4.97·56-s + 2.23·58-s + 1.17·59-s + 1.61·61-s + 2.61·64-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

\(L(\frac{1}{2})\) \(\approx\) \(2.558581974\)
\(L(\frac12)\) \(\approx\) \(2.558581974\)
\(L(1)\) 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 \( 1 - 1.90T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 0.618T + T^{2} \)
17 \( 1 + 1.17T + T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
23 \( 1 + 1.90T + T^{2} \)
29 \( 1 - 1.17T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.90T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.17T + T^{2} \)
61 \( 1 - 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 0.618T + T^{2} \)
83 \( 1 + 1.17T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.61T + T^{2} \)
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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

−10.33649315983170256938586077098, −9.855559264285757365894589158273, −8.460558806753381159575475572854, −7.09027976863346583663906770029, −6.65956504988089730504825770078, −5.89831921974029083967596286995, −4.92433245517473811817350748300, −3.98788792494418015169251516367, −3.13738861457169047996351408440, −2.27500541487441074042644019185, 2.27500541487441074042644019185, 3.13738861457169047996351408440, 3.98788792494418015169251516367, 4.92433245517473811817350748300, 5.89831921974029083967596286995, 6.65956504988089730504825770078, 7.09027976863346583663906770029, 8.460558806753381159575475572854, 9.855559264285757365894589158273, 10.33649315983170256938586077098

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