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Properties

Label 2-869-869.631-c0-0-3
Degree $2$
Conductor $869$
Sign $0.920 - 0.390i$
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  + (1.03 + 0.749i)2-s + (0.193 + 0.594i)4-s + (0.688 − 0.500i)5-s + (0.147 − 0.454i)8-s + (−0.809 − 0.587i)9-s + 1.08·10-s + (−0.637 − 0.770i)11-s + (1.50 + 1.09i)13-s + (0.998 − 0.725i)16-s + (−0.393 − 1.21i)18-s + (−0.613 + 1.88i)19-s + (0.430 + 0.312i)20-s + (−0.0800 − 1.27i)22-s − 1.98·23-s + (−0.0849 + 0.261i)25-s + (0.732 + 2.25i)26-s + ⋯
L(s)  = 1  + (1.03 + 0.749i)2-s + (0.193 + 0.594i)4-s + (0.688 − 0.500i)5-s + (0.147 − 0.454i)8-s + (−0.809 − 0.587i)9-s + 1.08·10-s + (−0.637 − 0.770i)11-s + (1.50 + 1.09i)13-s + (0.998 − 0.725i)16-s + (−0.393 − 1.21i)18-s + (−0.613 + 1.88i)19-s + (0.430 + 0.312i)20-s + (−0.0800 − 1.27i)22-s − 1.98·23-s + (−0.0849 + 0.261i)25-s + (0.732 + 2.25i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $0.920 - 0.390i$
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: \((2,\ 869,\ (\ :0),\ 0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.726548254\)
\(L(\frac12)\) \(\approx\) \(1.726548254\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.637 + 0.770i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.688 + 0.500i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + 1.98T + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 - 1.45T + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.115 + 0.356i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 - 1.07T + T^{2} \)
97 \( 1 + (1.41 + 1.03i)T + (0.309 + 0.951i)T^{2} \)
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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

−10.37118722998435784367838414964, −9.509399779997082835991608257300, −8.553577481959572853699626831680, −7.905198447511785103747461379193, −6.33778850051025756375283020801, −6.06607441121799044746782376549, −5.46611441349331111306570286775, −4.12792019078453533609195891885, −3.48982552125412581369036124290, −1.68405857144169826133496392287, 2.13345714951986106479017124540, 2.73231471970406633280676358466, 3.85105616171639722295000627008, 4.97047785185011739563691394019, 5.70028443769980476842239692869, 6.50975658204272010477775940336, 7.947291780369287548043979867764, 8.481539022291429932272750700673, 9.829549484337011601929875138011, 10.70847528863204648129128756891

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