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Properties

Label 2-873-97.20-c0-0-0
Degree $2$
Conductor $873$
Sign $0.507 + 0.861i$
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 + (−1.26 − 1.53i)7-s + (0.577 − 0.0569i)13-s + (0.707 − 0.707i)16-s + (−0.980 − 0.804i)19-s + (0.195 + 0.980i)25-s + (−1.75 − 0.938i)28-s + (1.81 + 0.360i)31-s + (0.273 + 0.902i)37-s + (−1.53 + 0.636i)43-s + (−0.577 + 2.90i)49-s + (0.512 − 0.273i)52-s + 1.11·61-s + (0.382 − 0.923i)64-s + (1.75 − 0.172i)67-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)4-s + (−1.26 − 1.53i)7-s + (0.577 − 0.0569i)13-s + (0.707 − 0.707i)16-s + (−0.980 − 0.804i)19-s + (0.195 + 0.980i)25-s + (−1.75 − 0.938i)28-s + (1.81 + 0.360i)31-s + (0.273 + 0.902i)37-s + (−1.53 + 0.636i)43-s + (−0.577 + 2.90i)49-s + (0.512 − 0.273i)52-s + 1.11·61-s + (0.382 − 0.923i)64-s + (1.75 − 0.172i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(873\)    =    \(3^{2} \cdot 97\)
Sign: $0.507 + 0.861i$
Analytic conductor: \(0.435683\)
Root analytic conductor: \(0.660063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{873} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 873,\ (\ :0),\ 0.507 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.071040098\)
\(L(\frac12)\) \(\approx\) \(1.071040098\)
\(L(1)\) 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 \( 1 + (1.26 + 1.53i)T + (-0.195 + 0.980i)T^{2} \)
11 \( 1 + (0.382 - 0.923i)T^{2} \)
13 \( 1 + (-0.577 + 0.0569i)T + (0.980 - 0.195i)T^{2} \)
17 \( 1 + (0.980 - 0.195i)T^{2} \)
19 \( 1 + (0.980 + 0.804i)T + (0.195 + 0.980i)T^{2} \)
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 \( 1 + (-0.273 - 0.902i)T + (-0.831 + 0.555i)T^{2} \)
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 \( 1 + (-1.75 + 0.172i)T + (0.980 - 0.195i)T^{2} \)
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

−10.19195523858017213903553164305, −9.746586247388889008646607649383, −8.478183681665661225552115293181, −7.42146841524000017658585850147, −6.60615544969255059741339509860, −6.33717256661825873531180497249, −4.85567047784267800598658768459, −3.68541780217451129048229375356, −2.80782177712783763677778499972, −1.13281357564073279559878800395, 2.13663072865797545152561821103, 2.91259011959098119757099618194, 3.94734023595445649953756861918, 5.56436124839264708472352333332, 6.31338779521247914582024949070, 6.74688842178574731615146001508, 8.219420207685552836373596272149, 8.599118505357002456523054333769, 9.755668409703781864443126412546, 10.40133370127814552694838890303

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