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Properties

Label 2-873-97.28-c0-0-0
Degree $2$
Conductor $873$
Sign $-0.763 - 0.645i$
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 + (−1.53 + 0.151i)7-s + (−1.36 + 1.11i)13-s + (0.707 + 0.707i)16-s + (−0.195 + 1.98i)19-s + (−0.980 − 0.195i)25-s + (1.47 + 0.448i)28-s + (−0.360 − 1.81i)31-s + (−0.512 + 0.273i)37-s + (−1.02 − 0.425i)43-s + (1.36 − 0.271i)49-s + (1.68 − 0.512i)52-s + 1.66·61-s + (−0.382 − 0.923i)64-s + (−1.47 + 1.21i)67-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)4-s + (−1.53 + 0.151i)7-s + (−1.36 + 1.11i)13-s + (0.707 + 0.707i)16-s + (−0.195 + 1.98i)19-s + (−0.980 − 0.195i)25-s + (1.47 + 0.448i)28-s + (−0.360 − 1.81i)31-s + (−0.512 + 0.273i)37-s + (−1.02 − 0.425i)43-s + (1.36 − 0.271i)49-s + (1.68 − 0.512i)52-s + 1.66·61-s + (−0.382 − 0.923i)64-s + (−1.47 + 1.21i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(873\)    =    \(3^{2} \cdot 97\)
Sign: $-0.763 - 0.645i$
Analytic conductor: \(0.435683\)
Root analytic conductor: \(0.660063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{873} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 873,\ (\ :0),\ -0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2125397486\)
\(L(\frac12)\) \(\approx\) \(0.2125397486\)
\(L(1)\) 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 \( 1 + (-0.980 - 0.195i)T \)
good2 \( 1 + (0.923 + 0.382i)T^{2} \)
5 \( 1 + (0.980 + 0.195i)T^{2} \)
7 \( 1 + (1.53 - 0.151i)T + (0.980 - 0.195i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + (1.36 - 1.11i)T + (0.195 - 0.980i)T^{2} \)
17 \( 1 + (0.195 - 0.980i)T^{2} \)
19 \( 1 + (0.195 - 1.98i)T + (-0.980 - 0.195i)T^{2} \)
23 \( 1 + (-0.831 + 0.555i)T^{2} \)
29 \( 1 + (-0.831 + 0.555i)T^{2} \)
31 \( 1 + (0.360 + 1.81i)T + (-0.923 + 0.382i)T^{2} \)
37 \( 1 + (0.512 - 0.273i)T + (0.555 - 0.831i)T^{2} \)
41 \( 1 + (-0.555 - 0.831i)T^{2} \)
43 \( 1 + (1.02 + 0.425i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.707 - 0.707i)T^{2} \)
53 \( 1 + (-0.382 + 0.923i)T^{2} \)
59 \( 1 + (0.831 + 0.555i)T^{2} \)
61 \( 1 - 1.66T + T^{2} \)
67 \( 1 + (1.47 - 1.21i)T + (0.195 - 0.980i)T^{2} \)
71 \( 1 + (-0.555 + 0.831i)T^{2} \)
73 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (-0.980 - 0.195i)T^{2} \)
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

−10.08890431620889633035207369243, −9.935995023666072883520336689182, −9.263008779136607142908487135433, −8.277806753824566035103278765757, −7.23670734037775123593983861309, −6.21929625244527889060375034973, −5.56430873864745207657184622083, −4.29998445325667135284533840258, −3.58567729077370727707418787866, −2.07197667636072683083826299033, 0.20354702442586224585966847381, 2.78438492725780323399008494488, 3.46493771294577492728922997813, 4.71915686322121831632833880386, 5.46922258372197594500944741112, 6.76336304941922998032374515559, 7.38918376935117380865850684575, 8.485030219831815406608017578859, 9.317194006745954875295885375377, 9.858136383514663232227378021254

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