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Properties

Label 2-873-97.51-c0-0-0
Degree $2$
Conductor $873$
Sign $0.365 + 0.930i$
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 + (−0.273 − 0.512i)7-s + (−0.368 − 1.21i)13-s + (−0.707 − 0.707i)16-s + (0.831 + 0.444i)19-s + (0.555 + 0.831i)25-s + (−0.577 + 0.0569i)28-s + (−0.636 − 0.425i)31-s + (−0.124 + 0.151i)37-s + (−0.149 + 0.360i)43-s + (0.368 − 0.551i)49-s + (−1.26 − 0.124i)52-s + 1.96·61-s + (−0.923 + 0.382i)64-s + (0.577 + 1.90i)67-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)4-s + (−0.273 − 0.512i)7-s + (−0.368 − 1.21i)13-s + (−0.707 − 0.707i)16-s + (0.831 + 0.444i)19-s + (0.555 + 0.831i)25-s + (−0.577 + 0.0569i)28-s + (−0.636 − 0.425i)31-s + (−0.124 + 0.151i)37-s + (−0.149 + 0.360i)43-s + (0.368 − 0.551i)49-s + (−1.26 − 0.124i)52-s + 1.96·61-s + (−0.923 + 0.382i)64-s + (0.577 + 1.90i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(873\)    =    \(3^{2} \cdot 97\)
Sign: $0.365 + 0.930i$
Analytic conductor: \(0.435683\)
Root analytic conductor: \(0.660063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{873} (730, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 873,\ (\ :0),\ 0.365 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.018469892\)
\(L(\frac12)\) \(\approx\) \(1.018469892\)
\(L(1)\) 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 \( 1 + (0.273 + 0.512i)T + (-0.555 + 0.831i)T^{2} \)
11 \( 1 + (-0.923 + 0.382i)T^{2} \)
13 \( 1 + (0.368 + 1.21i)T + (-0.831 + 0.555i)T^{2} \)
17 \( 1 + (-0.831 + 0.555i)T^{2} \)
19 \( 1 + (-0.831 - 0.444i)T + (0.555 + 0.831i)T^{2} \)
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 \( 1 + (0.124 - 0.151i)T + (-0.195 - 0.980i)T^{2} \)
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 \( 1 + (-0.577 - 1.90i)T + (-0.831 + 0.555i)T^{2} \)
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

−10.04950599861107623660388757026, −9.738288498651093064715393091153, −8.526348840240492526281255255344, −7.45234023959973090358652280460, −6.84535978930414181652426439109, −5.67085774088013484463768227375, −5.17865235856430098589798142289, −3.76230455103369578252794062026, −2.61053819148582862180905099096, −1.09589021186535098728036007809, 2.07391795022933394223123206073, 3.06252391166072613808843482298, 4.12436544557602630385827212521, 5.18101622075313897089303488381, 6.45833123203287710452371982079, 7.04642659076629494331115105094, 7.962245674575679156766029001897, 8.905606325513599950541765327401, 9.440667966860332298058807567768, 10.62638417322850817369429459624

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