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Properties

Label 2-873-97.45-c0-0-0
Degree $2$
Conductor $873$
Sign $0.756 + 0.654i$
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 + (0.124 − 1.26i)7-s + (0.598 − 0.728i)13-s + (0.707 − 0.707i)16-s + (0.195 − 0.0192i)19-s + (0.980 − 0.195i)25-s + (0.368 + 1.21i)28-s + (0.360 − 1.81i)31-s + (−0.902 + 1.68i)37-s + (1.02 − 0.425i)43-s + (−0.598 − 0.118i)49-s + (−0.273 + 0.902i)52-s − 1.66·61-s + (−0.382 + 0.923i)64-s + (−0.368 + 0.448i)67-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)4-s + (0.124 − 1.26i)7-s + (0.598 − 0.728i)13-s + (0.707 − 0.707i)16-s + (0.195 − 0.0192i)19-s + (0.980 − 0.195i)25-s + (0.368 + 1.21i)28-s + (0.360 − 1.81i)31-s + (−0.902 + 1.68i)37-s + (1.02 − 0.425i)43-s + (−0.598 − 0.118i)49-s + (−0.273 + 0.902i)52-s − 1.66·61-s + (−0.382 + 0.923i)64-s + (−0.368 + 0.448i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(873\)    =    \(3^{2} \cdot 97\)
Sign: $0.756 + 0.654i$
Analytic conductor: \(0.435683\)
Root analytic conductor: \(0.660063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{873} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 873,\ (\ :0),\ 0.756 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8209050560\)
\(L(\frac12)\) \(\approx\) \(0.8209050560\)
\(L(1)\) 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 \( 1 + (-0.124 + 1.26i)T + (-0.980 - 0.195i)T^{2} \)
11 \( 1 + (-0.382 + 0.923i)T^{2} \)
13 \( 1 + (-0.598 + 0.728i)T + (-0.195 - 0.980i)T^{2} \)
17 \( 1 + (-0.195 - 0.980i)T^{2} \)
19 \( 1 + (-0.195 + 0.0192i)T + (0.980 - 0.195i)T^{2} \)
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 \( 1 + (0.902 - 1.68i)T + (-0.555 - 0.831i)T^{2} \)
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 \( 1 + (0.368 - 0.448i)T + (-0.195 - 0.980i)T^{2} \)
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

−10.25545272765397357057457573941, −9.457140770383757626940236651630, −8.466917622410744663092632378814, −7.84427555306562669046979763435, −7.00377511962789305988270910199, −5.82529427379462984105019607742, −4.72663827227914861615534012566, −3.99715297040677393374645373251, −3.04085348357371223336700612268, −0.970883082900715143615229219429, 1.59625980175788511168599768024, 3.06812456679790437792603442197, 4.29711389761063457562292824925, 5.22642180848369340112686166565, 5.90755844380015762193843041874, 6.94325637765819832874040881144, 8.237658819577394737382575701919, 8.977282408782939949356990309113, 9.262032901072379720823000631565, 10.45567797651229878373812095644

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