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Properties

Label 2-873-97.55-c0-0-0
Degree $2$
Conductor $873$
Sign $0.537 - 0.843i$
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 + (0.902 + 0.273i)7-s + (0.0924 − 0.172i)13-s + (−0.707 − 0.707i)16-s + (0.555 + 1.83i)19-s + (−0.831 + 0.555i)25-s + (−0.598 + 0.728i)28-s + (0.425 − 0.636i)31-s + (0.151 + 1.53i)37-s + (0.750 − 1.81i)43-s + (−0.0924 − 0.0617i)49-s + (0.124 + 0.151i)52-s − 0.390·61-s + (0.923 − 0.382i)64-s + (0.598 − 1.11i)67-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)4-s + (0.902 + 0.273i)7-s + (0.0924 − 0.172i)13-s + (−0.707 − 0.707i)16-s + (0.555 + 1.83i)19-s + (−0.831 + 0.555i)25-s + (−0.598 + 0.728i)28-s + (0.425 − 0.636i)31-s + (0.151 + 1.53i)37-s + (0.750 − 1.81i)43-s + (−0.0924 − 0.0617i)49-s + (0.124 + 0.151i)52-s − 0.390·61-s + (0.923 − 0.382i)64-s + (0.598 − 1.11i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(873\)    =    \(3^{2} \cdot 97\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(0.435683\)
Root analytic conductor: \(0.660063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{873} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 873,\ (\ :0),\ 0.537 - 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9798713644\)
\(L(\frac12)\) \(\approx\) \(0.9798713644\)
\(L(1)\) 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 \( 1 + (-0.902 - 0.273i)T + (0.831 + 0.555i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (-0.0924 + 0.172i)T + (-0.555 - 0.831i)T^{2} \)
17 \( 1 + (-0.555 - 0.831i)T^{2} \)
19 \( 1 + (-0.555 - 1.83i)T + (-0.831 + 0.555i)T^{2} \)
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 \( 1 + (-0.151 - 1.53i)T + (-0.980 + 0.195i)T^{2} \)
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 \( 1 + (-0.598 + 1.11i)T + (-0.555 - 0.831i)T^{2} \)
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

−10.43480694849493795732690981225, −9.588123899548713627488281419015, −8.627508755652624306170150937377, −7.966973047444230083264287198905, −7.43008484885125094321263512277, −6.06702547414921362192906396019, −5.13503809969469963380337395480, −4.13119576474733590803279522410, −3.23201999854355149928455524051, −1.81914376239551131050008827839, 1.14183327764319155809840419064, 2.50836230081696021048207216122, 4.18851211773814009075201389899, 4.86696564868077685062362119793, 5.74423385069250463195904621756, 6.75694270595996601169649034860, 7.68518623892186266259296257962, 8.687342838256503726856866832431, 9.396063569383240564797222301923, 10.19830120552662349694233524735

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