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Properties

Label 2-867-51.14-c1-0-9
Degree $2$
Conductor $867$
Sign $-0.939 + 0.343i$
Analytic cond. $6.92302$
Root an. cond. $2.63116$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.762 + 1.84i)2-s + (−0.908 + 1.47i)3-s + (−1.39 + 1.39i)4-s + (0.240 + 0.360i)5-s + (−3.40 − 0.548i)6-s + (2.28 + 1.52i)7-s + (0.0550 + 0.0227i)8-s + (−1.34 − 2.68i)9-s + (−0.479 + 0.717i)10-s + (1.11 + 5.59i)11-s + (−0.787 − 3.32i)12-s + (−1.33 − 1.33i)13-s + (−1.07 + 5.38i)14-s + (−0.749 + 0.0274i)15-s + 4.05i·16-s + ⋯
L(s)  = 1  + (0.539 + 1.30i)2-s + (−0.524 + 0.851i)3-s + (−0.696 + 0.696i)4-s + (0.107 + 0.161i)5-s + (−1.39 − 0.224i)6-s + (0.865 + 0.578i)7-s + (0.0194 + 0.00805i)8-s + (−0.449 − 0.893i)9-s + (−0.151 + 0.226i)10-s + (0.335 + 1.68i)11-s + (−0.227 − 0.958i)12-s + (−0.370 − 0.370i)13-s + (−0.286 + 1.43i)14-s + (−0.193 + 0.00709i)15-s + 1.01i·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.939 + 0.343i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.939 + 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327752 - 1.84830i\)
\(L(\frac12)\) \(\approx\) \(0.327752 - 1.84830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.908 - 1.47i)T \)
17 \( 1 \)
good2 \( 1 + (-0.762 - 1.84i)T + (-1.41 + 1.41i)T^{2} \)
5 \( 1 + (-0.240 - 0.360i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (-2.28 - 1.52i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (-1.11 - 5.59i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (1.33 + 1.33i)T + 13iT^{2} \)
19 \( 1 + (-0.697 + 0.288i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.79 - 0.356i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (4.78 - 3.19i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (-0.0525 - 0.0104i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (-1.69 + 8.50i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (0.901 - 1.34i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (-3.22 - 1.33i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (1.44 - 1.44i)T - 47iT^{2} \)
53 \( 1 + (2.67 + 6.46i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-8.58 - 3.55i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-5.17 + 7.75i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + 2.50iT - 67T^{2} \)
71 \( 1 + (6.36 + 1.26i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-1.17 + 0.785i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (9.53 - 1.89i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (-7.32 + 3.03i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-7.83 - 7.83i)T + 89iT^{2} \)
97 \( 1 + (-6.22 - 9.31i)T + (-37.1 + 89.6i)T^{2} \)
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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

−10.55160861528440635989327546563, −9.732726250684388879640802776998, −8.858726896725777872385329241827, −7.84789450469064775924128561889, −7.06100433128083733591860831157, −6.18020308044217237285314769239, −5.25281896209551370172775854116, −4.79450288857090265096899836832, −3.91179494937537780103181452681, −2.09741391250212918900307743733, 0.862368489326040013295703281614, 1.75873485496083625005352589101, 3.00205005144931347204104038702, 4.12752675317621166230193501598, 5.11393062485871298042192958955, 5.97342528969205618603039451697, 7.15092909793040482725883255551, 7.954048659611506076684614146699, 8.906292506123807091139693898920, 10.12599829779576304375004113697

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