Properties

Label 2-869-869.236-c0-0-3
Degree $2$
Conductor $869$
Sign $-0.0798 - 0.996i$
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  + (0.541 + 1.66i)2-s + (−1.67 + 1.21i)4-s + (0.598 − 1.84i)5-s + (−1.51 − 1.10i)8-s + (0.309 + 0.951i)9-s + 3.39·10-s + (0.876 + 0.481i)11-s + (−0.263 − 0.809i)13-s + (0.377 − 1.16i)16-s + (−1.41 + 1.03i)18-s + (1.50 + 1.09i)19-s + (1.24 + 3.81i)20-s + (−0.328 + 1.72i)22-s − 1.85·23-s + (−2.22 − 1.61i)25-s + (1.20 − 0.877i)26-s + ⋯
L(s)  = 1  + (0.541 + 1.66i)2-s + (−1.67 + 1.21i)4-s + (0.598 − 1.84i)5-s + (−1.51 − 1.10i)8-s + (0.309 + 0.951i)9-s + 3.39·10-s + (0.876 + 0.481i)11-s + (−0.263 − 0.809i)13-s + (0.377 − 1.16i)16-s + (−1.41 + 1.03i)18-s + (1.50 + 1.09i)19-s + (1.24 + 3.81i)20-s + (−0.328 + 1.72i)22-s − 1.85·23-s + (−2.22 − 1.61i)25-s + (1.20 − 0.877i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0798 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0798 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $-0.0798 - 0.996i$
Analytic conductor: \(0.433687\)
Root analytic conductor: \(0.658549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{869} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 869,\ (\ :0),\ -0.0798 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346680345\)
\(L(\frac12)\) \(\approx\) \(1.346680345\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.876 - 0.481i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.541 - 1.66i)T + (-0.809 + 0.587i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + 1.85T + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.115 + 0.356i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + 1.27T + T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + 1.98T + T^{2} \)
97 \( 1 + (-0.0388 - 0.119i)T + (-0.809 + 0.587i)T^{2} \)
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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

−10.02658589870736109183095816347, −9.571717396128133845660847300879, −8.488083153939632112703744712056, −8.008010041805699104317797654101, −7.26186887539767158518470965761, −5.84487862237283949650916682059, −5.56067774194717251475956500841, −4.66270863282930419527159743404, −3.97372770609376117759918627080, −1.66998238131836381551886017037, 1.60163181060002706234892674355, 2.73526222556662928273110341323, 3.44774920933943332765024053498, 4.24045470478494384940280440715, 5.73653217622183876146735468055, 6.52875503996804492650855853694, 7.34807217927306274058153958827, 9.178241516124247872750125965563, 9.595322356291221475251135583574, 10.28474129034656348575270592785

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