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Properties

Label 2-867-17.13-c1-0-3
Degree $2$
Conductor $867$
Sign $-0.698 - 0.715i$
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  + 0.765i·2-s + (−0.707 − 0.707i)3-s + 1.41·4-s + (−1.47 − 1.47i)5-s + (0.541 − 0.541i)6-s + (−0.873 + 0.873i)7-s + 2.61i·8-s + 1.00i·9-s + (1.12 − 1.12i)10-s + (−1.05 + 1.05i)11-s + (−1 − i)12-s − 4.10·13-s + (−0.668 − 0.668i)14-s + 2.08i·15-s + 0.828·16-s + ⋯
L(s)  = 1  + 0.541i·2-s + (−0.408 − 0.408i)3-s + 0.707·4-s + (−0.658 − 0.658i)5-s + (0.220 − 0.220i)6-s + (−0.329 + 0.329i)7-s + 0.923i·8-s + 0.333i·9-s + (0.356 − 0.356i)10-s + (−0.319 + 0.319i)11-s + (−0.288 − 0.288i)12-s − 1.13·13-s + (−0.178 − 0.178i)14-s + 0.537i·15-s + 0.207·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.698 - 0.715i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.698 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.284072 + 0.673872i\)
\(L(\frac12)\) \(\approx\) \(0.284072 + 0.673872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 \)
good2 \( 1 - 0.765iT - 2T^{2} \)
5 \( 1 + (1.47 + 1.47i)T + 5iT^{2} \)
7 \( 1 + (0.873 - 0.873i)T - 7iT^{2} \)
11 \( 1 + (1.05 - 1.05i)T - 11iT^{2} \)
13 \( 1 + 4.10T + 13T^{2} \)
19 \( 1 - 6.81iT - 19T^{2} \)
23 \( 1 + (2.33 - 2.33i)T - 23iT^{2} \)
29 \( 1 + (4.41 + 4.41i)T + 29iT^{2} \)
31 \( 1 + (-7.26 - 7.26i)T + 31iT^{2} \)
37 \( 1 + (-2.78 - 2.78i)T + 37iT^{2} \)
41 \( 1 + (8.58 - 8.58i)T - 41iT^{2} \)
43 \( 1 + 3.44iT - 43T^{2} \)
47 \( 1 + 1.56T + 47T^{2} \)
53 \( 1 + 3.96iT - 53T^{2} \)
59 \( 1 - 8.06iT - 59T^{2} \)
61 \( 1 + (2.31 - 2.31i)T - 61iT^{2} \)
67 \( 1 - 2.11T + 67T^{2} \)
71 \( 1 + (0.160 + 0.160i)T + 71iT^{2} \)
73 \( 1 + (-0.260 - 0.260i)T + 73iT^{2} \)
79 \( 1 + (-1.91 + 1.91i)T - 79iT^{2} \)
83 \( 1 + 14.4iT - 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + (-1.89 - 1.89i)T + 97iT^{2} \)
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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

−10.39330466488988905520710745594, −9.747271590183216437399809424335, −8.314785522943871067213915036786, −7.922386803258311355871247827922, −7.08311791756847474468003269871, −6.19096715460338568892331027803, −5.39360299503949377032499194302, −4.45485825137561622501774813878, −2.94106880506974123843270162396, −1.68553165213893966798130217966, 0.34469708449368383128927626178, 2.39046425496300314744306336667, 3.24509184002893691065985375533, 4.21732114822538089347976161966, 5.37327807468243160202990941487, 6.63666997557011847597717689870, 7.08419499683838549629945153686, 7.989821708466036778911753980248, 9.381328178760191970802190049665, 10.06209716276723049727049783248

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