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Properties

Label 2-873-97.46-c0-0-0
Degree $2$
Conductor $873$
Sign $0.637 + 0.770i$
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 + (1.68 − 0.902i)7-s + (−1.47 + 0.448i)13-s + (−0.707 − 0.707i)16-s + (−0.831 + 1.55i)19-s + (−0.555 − 0.831i)25-s + (−0.187 − 1.90i)28-s + (0.636 + 0.425i)31-s + (1.53 + 1.26i)37-s + (0.149 − 0.360i)43-s + (1.47 − 2.21i)49-s + (−0.151 + 1.53i)52-s − 1.96·61-s + (−0.923 + 0.382i)64-s + (0.187 − 0.0569i)67-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)4-s + (1.68 − 0.902i)7-s + (−1.47 + 0.448i)13-s + (−0.707 − 0.707i)16-s + (−0.831 + 1.55i)19-s + (−0.555 − 0.831i)25-s + (−0.187 − 1.90i)28-s + (0.636 + 0.425i)31-s + (1.53 + 1.26i)37-s + (0.149 − 0.360i)43-s + (1.47 − 2.21i)49-s + (−0.151 + 1.53i)52-s − 1.96·61-s + (−0.923 + 0.382i)64-s + (0.187 − 0.0569i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(873\)    =    \(3^{2} \cdot 97\)
Sign: $0.637 + 0.770i$
Analytic conductor: \(0.435683\)
Root analytic conductor: \(0.660063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{873} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 873,\ (\ :0),\ 0.637 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.174601651\)
\(L(\frac12)\) \(\approx\) \(1.174601651\)
\(L(1)\) 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 \( 1 + (-0.555 - 0.831i)T \)
good2 \( 1 + (-0.382 + 0.923i)T^{2} \)
5 \( 1 + (0.555 + 0.831i)T^{2} \)
7 \( 1 + (-1.68 + 0.902i)T + (0.555 - 0.831i)T^{2} \)
11 \( 1 + (-0.923 + 0.382i)T^{2} \)
13 \( 1 + (1.47 - 0.448i)T + (0.831 - 0.555i)T^{2} \)
17 \( 1 + (0.831 - 0.555i)T^{2} \)
19 \( 1 + (0.831 - 1.55i)T + (-0.555 - 0.831i)T^{2} \)
23 \( 1 + (0.980 + 0.195i)T^{2} \)
29 \( 1 + (0.980 + 0.195i)T^{2} \)
31 \( 1 + (-0.636 - 0.425i)T + (0.382 + 0.923i)T^{2} \)
37 \( 1 + (-1.53 - 1.26i)T + (0.195 + 0.980i)T^{2} \)
41 \( 1 + (-0.195 + 0.980i)T^{2} \)
43 \( 1 + (-0.149 + 0.360i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + (0.707 + 0.707i)T^{2} \)
53 \( 1 + (-0.923 - 0.382i)T^{2} \)
59 \( 1 + (-0.980 + 0.195i)T^{2} \)
61 \( 1 + 1.96T + T^{2} \)
67 \( 1 + (-0.187 + 0.0569i)T + (0.831 - 0.555i)T^{2} \)
71 \( 1 + (-0.195 - 0.980i)T^{2} \)
73 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.617 - 0.923i)T + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (-0.555 - 0.831i)T^{2} \)
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

−10.23089681101037697223022650146, −9.742061762025986235146106771191, −8.346091591860330697396174916271, −7.72502997015733101884135307442, −6.85022255975039753392156907492, −5.84653616396044699292887825970, −4.78144690721236767716156081201, −4.28859248049170599807089921917, −2.34220287757922849915631445119, −1.40097538325059712507504642898, 2.10709150590364125851547344004, 2.74930528932712349599109539927, 4.38822877483185302351965633723, 4.99366016905611748692114074335, 6.13793847489178212257811125486, 7.48463624856851664682602167948, 7.74084385723075848434965038593, 8.730408312712736030206437243384, 9.386653708552690928370765302291, 10.79113881037889470115347097677

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