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Properties

Label 2-867-17.4-c1-0-13
Degree $2$
Conductor $867$
Sign $-0.769 + 0.638i$
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  − 2.56i·2-s + (−0.707 + 0.707i)3-s − 4.56·4-s + (−2.51 + 2.51i)5-s + (1.81 + 1.81i)6-s + 6.56i·8-s − 1.00i·9-s + (6.45 + 6.45i)10-s + (1.10 + 1.10i)11-s + (3.22 − 3.22i)12-s − 0.438·13-s − 3.56i·15-s + 7.68·16-s − 2.56·18-s − 4.68i·19-s + (11.4 − 11.4i)20-s + ⋯
L(s)  = 1  − 1.81i·2-s + (−0.408 + 0.408i)3-s − 2.28·4-s + (−1.12 + 1.12i)5-s + (0.739 + 0.739i)6-s + 2.31i·8-s − 0.333i·9-s + (2.03 + 2.03i)10-s + (0.332 + 0.332i)11-s + (0.931 − 0.931i)12-s − 0.121·13-s − 0.919i·15-s + 1.92·16-s − 0.603·18-s − 1.07i·19-s + (2.56 − 2.56i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.769 + 0.638i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (616, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.769 + 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.234677 - 0.650629i\)
\(L(\frac12)\) \(\approx\) \(0.234677 - 0.650629i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.56iT - 2T^{2} \)
5 \( 1 + (2.51 - 2.51i)T - 5iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + (-1.10 - 1.10i)T + 11iT^{2} \)
13 \( 1 + 0.438T + 13T^{2} \)
19 \( 1 + 4.68iT - 19T^{2} \)
23 \( 1 + (1.72 + 1.72i)T + 23iT^{2} \)
29 \( 1 + (-5.83 + 5.83i)T - 29iT^{2} \)
31 \( 1 + (-2.20 + 2.20i)T - 31iT^{2} \)
37 \( 1 + (3.62 - 3.62i)T - 37iT^{2} \)
41 \( 1 + (-2.51 - 2.51i)T + 41iT^{2} \)
43 \( 1 + 4.68iT - 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + 7.12iT - 59T^{2} \)
61 \( 1 + (6.45 + 6.45i)T + 61iT^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (4.41 - 4.41i)T - 71iT^{2} \)
73 \( 1 + (-8.65 + 8.65i)T - 73iT^{2} \)
79 \( 1 + (6.62 + 6.62i)T + 79iT^{2} \)
83 \( 1 + 0.876iT - 83T^{2} \)
89 \( 1 - 1.12T + 89T^{2} \)
97 \( 1 + (-2.03 + 2.03i)T - 97iT^{2} \)
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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

−10.27595718248656123919272971601, −9.384456032136012424678672640055, −8.456681891130128462148994759276, −7.37669755000637804654643034350, −6.32277630543360084100494885085, −4.75215643082670859573917068400, −4.14388183719852805730221715034, −3.24689155830243058292836893154, −2.38151406051110860015053155716, −0.48891686718318926502720010347, 0.967853885553078349046446313846, 3.75546890601692633265261194999, 4.59641708936432862726687785699, 5.39651995696165934887864501239, 6.19460895575672582958585474783, 7.19677486758496505046984951569, 7.80308309259413540416904745801, 8.534041062564316455081618308663, 9.032422310397191948450248610550, 10.26667865362740682264396478291

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