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Properties

Label 2-927-103.89-c0-0-0
Degree $2$
Conductor $927$
Sign $0.822 - 0.568i$
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  + (0.273 + 0.961i)4-s + (0.831 − 0.322i)7-s + (0.510 − 0.197i)13-s + (−0.850 + 0.526i)16-s + (0.0822 − 0.165i)19-s + (−0.982 + 0.183i)25-s + (0.537 + 0.711i)28-s + (0.709 + 1.14i)31-s + (−0.486 − 0.533i)37-s + (0.247 − 0.271i)43-s + (−0.151 + 0.138i)49-s + (0.329 + 0.436i)52-s + (−1.67 + 0.312i)61-s + (−0.739 − 0.673i)64-s + (0.576 − 1.48i)67-s + ⋯
L(s)  = 1  + (0.273 + 0.961i)4-s + (0.831 − 0.322i)7-s + (0.510 − 0.197i)13-s + (−0.850 + 0.526i)16-s + (0.0822 − 0.165i)19-s + (−0.982 + 0.183i)25-s + (0.537 + 0.711i)28-s + (0.709 + 1.14i)31-s + (−0.486 − 0.533i)37-s + (0.247 − 0.271i)43-s + (−0.151 + 0.138i)49-s + (0.329 + 0.436i)52-s + (−1.67 + 0.312i)61-s + (−0.739 − 0.673i)64-s + (0.576 − 1.48i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(0.462633\)
Root analytic conductor: \(0.680171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :0),\ 0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.151332343\)
\(L(\frac12)\) \(\approx\) \(1.151332343\)
\(L(1)\) 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 \( 1 + (0.739 - 0.673i)T \)
good2 \( 1 + (-0.273 - 0.961i)T^{2} \)
5 \( 1 + (0.982 - 0.183i)T^{2} \)
7 \( 1 + (-0.831 + 0.322i)T + (0.739 - 0.673i)T^{2} \)
11 \( 1 + (0.273 + 0.961i)T^{2} \)
13 \( 1 + (-0.510 + 0.197i)T + (0.739 - 0.673i)T^{2} \)
17 \( 1 + (0.932 - 0.361i)T^{2} \)
19 \( 1 + (-0.0822 + 0.165i)T + (-0.602 - 0.798i)T^{2} \)
23 \( 1 + (-0.273 + 0.961i)T^{2} \)
29 \( 1 + (-0.982 + 0.183i)T^{2} \)
31 \( 1 + (-0.709 - 1.14i)T + (-0.445 + 0.895i)T^{2} \)
37 \( 1 + (0.486 + 0.533i)T + (-0.0922 + 0.995i)T^{2} \)
41 \( 1 + (-0.982 - 0.183i)T^{2} \)
43 \( 1 + (-0.247 + 0.271i)T + (-0.0922 - 0.995i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.602 + 0.798i)T^{2} \)
59 \( 1 + (0.739 + 0.673i)T^{2} \)
61 \( 1 + (1.67 - 0.312i)T + (0.932 - 0.361i)T^{2} \)
67 \( 1 + (-0.576 + 1.48i)T + (-0.739 - 0.673i)T^{2} \)
71 \( 1 + (0.982 + 0.183i)T^{2} \)
73 \( 1 + (-1.04 - 0.0971i)T + (0.982 + 0.183i)T^{2} \)
79 \( 1 + (0.172 + 1.85i)T + (-0.982 + 0.183i)T^{2} \)
83 \( 1 + (0.739 - 0.673i)T^{2} \)
89 \( 1 + (0.850 + 0.526i)T^{2} \)
97 \( 1 + (1.93 + 0.361i)T + (0.932 + 0.361i)T^{2} \)
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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

−10.60516402694113804968793344672, −9.392278487198149322767103566129, −8.493958778996007764745790395831, −7.87745731845981600239199440916, −7.14442239506516673021549167061, −6.15940306820200661440498092605, −4.95814480075655306836828347111, −4.02211483938149885535944082465, −3.06401643188822533943155535247, −1.73650375368086800303389775957, 1.41054860711271730454305805993, 2.46830257465253837004255807325, 4.05086882147440304872581536927, 5.04899453825730865616663847425, 5.85423707882778009760858611532, 6.61766962457309227290196809553, 7.73965709384074977541662008411, 8.515795981933753304901722737678, 9.508459674744259012471461269265, 10.14977423764164933889113620794

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