Properties

Label 2-869-869.631-c0-0-1
Degree $2$
Conductor $869$
Sign $-0.0864 - 0.996i$
Analytic cond. $0.433687$
Root an. cond. $0.658549$
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  + (1.50 + 1.09i)2-s + (0.759 + 2.33i)4-s + (0.303 − 0.220i)5-s + (−0.837 + 2.57i)8-s + (−0.809 − 0.587i)9-s + 0.696·10-s + (−0.929 + 0.368i)11-s + (−0.101 − 0.0738i)13-s + (−2.09 + 1.51i)16-s + (−0.574 − 1.76i)18-s + (0.541 − 1.66i)19-s + (0.745 + 0.541i)20-s + (−1.80 − 0.462i)22-s + 1.75·23-s + (−0.265 + 0.817i)25-s + (−0.0721 − 0.222i)26-s + ⋯
L(s)  = 1  + (1.50 + 1.09i)2-s + (0.759 + 2.33i)4-s + (0.303 − 0.220i)5-s + (−0.837 + 2.57i)8-s + (−0.809 − 0.587i)9-s + 0.696·10-s + (−0.929 + 0.368i)11-s + (−0.101 − 0.0738i)13-s + (−2.09 + 1.51i)16-s + (−0.574 − 1.76i)18-s + (0.541 − 1.66i)19-s + (0.745 + 0.541i)20-s + (−1.80 − 0.462i)22-s + 1.75·23-s + (−0.265 + 0.817i)25-s + (−0.0721 − 0.222i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $-0.0864 - 0.996i$
Analytic conductor: \(0.433687\)
Root analytic conductor: \(0.658549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{869} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 869,\ (\ :0),\ -0.0864 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.070771033\)
\(L(\frac12)\) \(\approx\) \(2.070771033\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.929 - 0.368i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.303 + 0.220i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - 1.75T + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (1.56 + 1.13i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.98T + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + 1.27T + T^{2} \)
97 \( 1 + (-0.688 - 0.500i)T + (0.309 + 0.951i)T^{2} \)
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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

−11.02428468453140265514496746910, −9.411891648321095109960939390888, −8.757895607395666125557422206707, −7.55674184107818684607840633335, −7.09825669319460161664264086280, −6.02353261204997811092190208082, −5.29786896521867999543026712822, −4.72127230804725142268392633601, −3.38787984440629311350627760376, −2.62749725370649178838124449622, 1.74414356698575704406689491610, 2.83823165263871363299504520365, 3.49127957256481496498483528100, 4.86302319880267332169176033174, 5.45660386630276830983462467576, 6.15089465284597564583483868252, 7.39710607492159258581359783290, 8.573037862198494881396160451612, 9.759731170792491884811146737136, 10.61332718369721933352943670062

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