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Properties

Label 2-873-97.67-c0-0-0
Degree $2$
Conductor $873$
Sign $0.992 + 0.125i$
Analytic cond. $0.435683$
Root an. cond. $0.660063$
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  + (−0.382 − 0.923i)4-s + (0.512 + 1.68i)7-s + (1.75 − 0.938i)13-s + (−0.707 + 0.707i)16-s + (−0.555 − 0.168i)19-s + (0.831 + 0.555i)25-s + (1.36 − 1.11i)28-s + (−0.425 − 0.636i)31-s + (1.26 + 0.124i)37-s + (−0.750 − 1.81i)43-s + (−1.75 + 1.17i)49-s + (−1.53 − 1.26i)52-s + 0.390·61-s + (0.923 + 0.382i)64-s + (−1.36 + 0.728i)67-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)4-s + (0.512 + 1.68i)7-s + (1.75 − 0.938i)13-s + (−0.707 + 0.707i)16-s + (−0.555 − 0.168i)19-s + (0.831 + 0.555i)25-s + (1.36 − 1.11i)28-s + (−0.425 − 0.636i)31-s + (1.26 + 0.124i)37-s + (−0.750 − 1.81i)43-s + (−1.75 + 1.17i)49-s + (−1.53 − 1.26i)52-s + 0.390·61-s + (0.923 + 0.382i)64-s + (−1.36 + 0.728i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(873\)    =    \(3^{2} \cdot 97\)
Sign: $0.992 + 0.125i$
Analytic conductor: \(0.435683\)
Root analytic conductor: \(0.660063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{873} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 873,\ (\ :0),\ 0.992 + 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.005097578\)
\(L(\frac12)\) \(\approx\) \(1.005097578\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
97 \( 1 + (0.831 + 0.555i)T \)
good2 \( 1 + (0.382 + 0.923i)T^{2} \)
5 \( 1 + (-0.831 - 0.555i)T^{2} \)
7 \( 1 + (-0.512 - 1.68i)T + (-0.831 + 0.555i)T^{2} \)
11 \( 1 + (0.923 + 0.382i)T^{2} \)
13 \( 1 + (-1.75 + 0.938i)T + (0.555 - 0.831i)T^{2} \)
17 \( 1 + (0.555 - 0.831i)T^{2} \)
19 \( 1 + (0.555 + 0.168i)T + (0.831 + 0.555i)T^{2} \)
23 \( 1 + (-0.195 - 0.980i)T^{2} \)
29 \( 1 + (-0.195 - 0.980i)T^{2} \)
31 \( 1 + (0.425 + 0.636i)T + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (-1.26 - 0.124i)T + (0.980 + 0.195i)T^{2} \)
41 \( 1 + (-0.980 + 0.195i)T^{2} \)
43 \( 1 + (0.750 + 1.81i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (0.707 - 0.707i)T^{2} \)
53 \( 1 + (0.923 - 0.382i)T^{2} \)
59 \( 1 + (0.195 - 0.980i)T^{2} \)
61 \( 1 - 0.390T + T^{2} \)
67 \( 1 + (1.36 - 0.728i)T + (0.555 - 0.831i)T^{2} \)
71 \( 1 + (-0.980 - 0.195i)T^{2} \)
73 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (1.38 - 0.923i)T + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.831 + 0.555i)T^{2} \)
89 \( 1 + (0.923 + 0.382i)T^{2} \)
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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

−10.42816673105570390993767695673, −9.321858092985768791512217216612, −8.706850178703898246792894336126, −8.186097761634838803840949799481, −6.63759712545864211331050016526, −5.66717758784255316757055436172, −5.44282363684917684093501489168, −4.13213055910846582875993092919, −2.71594477581048078577143756092, −1.45400868270340261821843340853, 1.37977143531353131221302993029, 3.22753761273288580978989894506, 4.15238053351800324052620655950, 4.59557452814816751643021573450, 6.28705739692050330718031684173, 7.02926873842544283602297488966, 7.936938684830098345989415099784, 8.530297719866428342538471802465, 9.419596276735963311193544661159, 10.58581965029455511423002161501

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