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Properties

Label 2-873-97.77-c0-0-0
Degree $2$
Conductor $873$
Sign $0.994 + 0.105i$
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  + (0.923 − 0.382i)4-s + (−0.151 + 0.124i)7-s + (0.187 + 1.90i)13-s + (0.707 − 0.707i)16-s + (0.980 − 1.19i)19-s + (−0.195 − 0.980i)25-s + (−0.0924 + 0.172i)28-s + (−1.81 − 0.360i)31-s + (−1.68 + 0.512i)37-s + (1.53 − 0.636i)43-s + (−0.187 + 0.943i)49-s + (0.902 + 1.68i)52-s − 1.11·61-s + (0.382 − 0.923i)64-s + (0.0924 + 0.938i)67-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)4-s + (−0.151 + 0.124i)7-s + (0.187 + 1.90i)13-s + (0.707 − 0.707i)16-s + (0.980 − 1.19i)19-s + (−0.195 − 0.980i)25-s + (−0.0924 + 0.172i)28-s + (−1.81 − 0.360i)31-s + (−1.68 + 0.512i)37-s + (1.53 − 0.636i)43-s + (−0.187 + 0.943i)49-s + (0.902 + 1.68i)52-s − 1.11·61-s + (0.382 − 0.923i)64-s + (0.0924 + 0.938i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(873\)    =    \(3^{2} \cdot 97\)
Sign: $0.994 + 0.105i$
Analytic conductor: \(0.435683\)
Root analytic conductor: \(0.660063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{873} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 873,\ (\ :0),\ 0.994 + 0.105i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.204342752\)
\(L(\frac12)\) \(\approx\) \(1.204342752\)
\(L(1)\) 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 \( 1 + (-0.195 - 0.980i)T \)
good2 \( 1 + (-0.923 + 0.382i)T^{2} \)
5 \( 1 + (0.195 + 0.980i)T^{2} \)
7 \( 1 + (0.151 - 0.124i)T + (0.195 - 0.980i)T^{2} \)
11 \( 1 + (0.382 - 0.923i)T^{2} \)
13 \( 1 + (-0.187 - 1.90i)T + (-0.980 + 0.195i)T^{2} \)
17 \( 1 + (-0.980 + 0.195i)T^{2} \)
19 \( 1 + (-0.980 + 1.19i)T + (-0.195 - 0.980i)T^{2} \)
23 \( 1 + (0.555 - 0.831i)T^{2} \)
29 \( 1 + (0.555 - 0.831i)T^{2} \)
31 \( 1 + (1.81 + 0.360i)T + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (1.68 - 0.512i)T + (0.831 - 0.555i)T^{2} \)
41 \( 1 + (-0.831 - 0.555i)T^{2} \)
43 \( 1 + (-1.53 + 0.636i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T^{2} \)
53 \( 1 + (0.382 + 0.923i)T^{2} \)
59 \( 1 + (-0.555 - 0.831i)T^{2} \)
61 \( 1 + 1.11T + T^{2} \)
67 \( 1 + (-0.0924 - 0.938i)T + (-0.980 + 0.195i)T^{2} \)
71 \( 1 + (-0.831 + 0.555i)T^{2} \)
73 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \)
83 \( 1 + (-0.195 - 0.980i)T^{2} \)
89 \( 1 + (0.382 - 0.923i)T^{2} \)
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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

−10.45904173449402157176518769561, −9.398590001823289948068903182541, −8.920849458009513805792496027403, −7.50211422259609699698411434985, −6.92978343720758506335192809459, −6.14205844630584093273098394026, −5.14527784822503760496006445481, −3.98780686929913879824389266548, −2.68894286356928013922568440408, −1.64198655457169020702685610126, 1.61794798135819418251357889077, 3.10549708334884939625537132511, 3.63875266226716809648642164017, 5.41606835560463997715469490642, 5.88026625584495693769162982017, 7.22579292020302988414741900162, 7.62874648912401323185710726768, 8.527370503370471261799377595307, 9.657505905218682326089914629160, 10.57458990402809121410200915273

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