Properties

Label 2-927-103.95-c0-0-0
Degree $2$
Conductor $927$
Sign $0.705 - 0.708i$
Analytic cond. $0.462633$
Root an. cond. $0.680171$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.932 + 0.361i)4-s + (−0.156 − 0.0971i)7-s + (1.58 + 0.981i)13-s + (0.739 − 0.673i)16-s + (−0.111 + 1.20i)19-s + (−0.273 + 0.961i)25-s + (0.181 + 0.0339i)28-s + (1.20 + 1.32i)31-s + (0.942 − 0.469i)37-s + (−1.72 − 0.857i)43-s + (−0.430 − 0.864i)49-s + (−1.83 − 0.342i)52-s + (0.404 − 1.42i)61-s + (−0.445 + 0.895i)64-s + (0.193 + 0.312i)67-s + ⋯
L(s)  = 1  + (−0.932 + 0.361i)4-s + (−0.156 − 0.0971i)7-s + (1.58 + 0.981i)13-s + (0.739 − 0.673i)16-s + (−0.111 + 1.20i)19-s + (−0.273 + 0.961i)25-s + (0.181 + 0.0339i)28-s + (1.20 + 1.32i)31-s + (0.942 − 0.469i)37-s + (−1.72 − 0.857i)43-s + (−0.430 − 0.864i)49-s + (−1.83 − 0.342i)52-s + (0.404 − 1.42i)61-s + (−0.445 + 0.895i)64-s + (0.193 + 0.312i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.705 - 0.708i$
Analytic conductor: \(0.462633\)
Root analytic conductor: \(0.680171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :0),\ 0.705 - 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8298119426\)
\(L(\frac12)\) \(\approx\) \(0.8298119426\)
\(L(1)\) 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 \( 1 + (0.445 + 0.895i)T \)
good2 \( 1 + (0.932 - 0.361i)T^{2} \)
5 \( 1 + (0.273 - 0.961i)T^{2} \)
7 \( 1 + (0.156 + 0.0971i)T + (0.445 + 0.895i)T^{2} \)
11 \( 1 + (-0.932 + 0.361i)T^{2} \)
13 \( 1 + (-1.58 - 0.981i)T + (0.445 + 0.895i)T^{2} \)
17 \( 1 + (-0.850 - 0.526i)T^{2} \)
19 \( 1 + (0.111 - 1.20i)T + (-0.982 - 0.183i)T^{2} \)
23 \( 1 + (0.932 + 0.361i)T^{2} \)
29 \( 1 + (-0.273 + 0.961i)T^{2} \)
31 \( 1 + (-1.20 - 1.32i)T + (-0.0922 + 0.995i)T^{2} \)
37 \( 1 + (-0.942 + 0.469i)T + (0.602 - 0.798i)T^{2} \)
41 \( 1 + (-0.273 - 0.961i)T^{2} \)
43 \( 1 + (1.72 + 0.857i)T + (0.602 + 0.798i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.982 + 0.183i)T^{2} \)
59 \( 1 + (0.445 - 0.895i)T^{2} \)
61 \( 1 + (-0.404 + 1.42i)T + (-0.850 - 0.526i)T^{2} \)
67 \( 1 + (-0.193 - 0.312i)T + (-0.445 + 0.895i)T^{2} \)
71 \( 1 + (0.273 + 0.961i)T^{2} \)
73 \( 1 + (-1.07 - 0.811i)T + (0.273 + 0.961i)T^{2} \)
79 \( 1 + (1.02 + 1.35i)T + (-0.273 + 0.961i)T^{2} \)
83 \( 1 + (0.445 + 0.895i)T^{2} \)
89 \( 1 + (-0.739 - 0.673i)T^{2} \)
97 \( 1 + (0.149 + 0.526i)T + (-0.850 + 0.526i)T^{2} \)
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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

−10.21925301312595434758261576933, −9.505311080923600599387497269632, −8.592645661437697591709884519550, −8.181526287635128282842626180718, −6.96309192607546030467932270181, −6.06765146263616835040649015434, −5.04767693999324058500847702869, −3.97295306323785208629617613051, −3.38049933342539756970711206691, −1.51769311471812165477079952874, 0.980549395490329834323933257357, 2.82331257116847354831222449366, 3.97844203097580855679799707512, 4.83519450883020716948212381789, 5.89438121522051994619245398503, 6.48628228586436703678799008587, 8.024965789189925357515082914831, 8.416081590395662980686403116191, 9.389625333909602306566853740904, 10.06899291454981678795289174480

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