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Properties

Label 2-867-17.16-c1-0-30
Degree $2$
Conductor $867$
Sign $0.970 - 0.242i$
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  + 2.56·2-s − i·3-s + 4.56·4-s + 3.56i·5-s − 2.56i·6-s + 6.56·8-s − 9-s + 9.12i·10-s − 1.56i·11-s − 4.56i·12-s + 0.438·13-s + 3.56·15-s + 7.68·16-s − 2.56·18-s + 4.68·19-s + 16.2i·20-s + ⋯
L(s)  = 1  + 1.81·2-s − 0.577i·3-s + 2.28·4-s + 1.59i·5-s − 1.04i·6-s + 2.31·8-s − 0.333·9-s + 2.88i·10-s − 0.470i·11-s − 1.31i·12-s + 0.121·13-s + 0.919·15-s + 1.92·16-s − 0.603·18-s + 1.07·19-s + 3.63i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.51763 + 0.556145i\)
\(L(\frac12)\) \(\approx\) \(4.51763 + 0.556145i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
17 \( 1 \)
good2 \( 1 - 2.56T + 2T^{2} \)
5 \( 1 - 3.56iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 1.56iT - 11T^{2} \)
13 \( 1 - 0.438T + 13T^{2} \)
19 \( 1 - 4.68T + 19T^{2} \)
23 \( 1 - 2.43iT - 23T^{2} \)
29 \( 1 + 8.24iT - 29T^{2} \)
31 \( 1 - 3.12iT - 31T^{2} \)
37 \( 1 + 5.12iT - 37T^{2} \)
41 \( 1 - 3.56iT - 41T^{2} \)
43 \( 1 + 4.68T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 + 9.12iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 6.24iT - 71T^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 - 9.36iT - 79T^{2} \)
83 \( 1 - 0.876T + 83T^{2} \)
89 \( 1 + 1.12T + 89T^{2} \)
97 \( 1 + 2.87iT - 97T^{2} \)
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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

−10.68454658913161547821531802662, −9.632120482146273183543343680100, −7.920038894781074164294886327524, −7.28724249778668748594292780669, −6.41562075908726301516969512713, −5.99941058863323363928885223133, −4.90562790005371530818292158063, −3.52201291322088320789591099123, −3.07996250754678394068736016396, −1.99375384886021617023743166884, 1.54914932149894301646994621896, 3.05457188830752067390363115608, 4.05169951256191581691730623027, 4.90197516509079343255487019402, 5.21004957894356305996809834004, 6.23076007081288866423770318278, 7.34385293767476199304667873536, 8.425564041417115729344720382012, 9.338203878433493366681596027928, 10.27610248738954518565662347246

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