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Properties

Label 2-867-17.13-c1-0-40
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  − 1.56i·2-s + (−0.707 − 0.707i)3-s − 0.438·4-s + (0.397 + 0.397i)5-s + (−1.10 + 1.10i)6-s − 2.43i·8-s + 1.00i·9-s + (0.620 − 0.620i)10-s + (−1.81 + 1.81i)11-s + (0.310 + 0.310i)12-s − 4.56·13-s − 0.561i·15-s − 4.68·16-s + 1.56·18-s − 7.68i·19-s + (−0.174 − 0.174i)20-s + ⋯
L(s)  = 1  − 1.10i·2-s + (−0.408 − 0.408i)3-s − 0.219·4-s + (0.177 + 0.177i)5-s + (−0.450 + 0.450i)6-s − 0.862i·8-s + 0.333i·9-s + (0.196 − 0.196i)10-s + (−0.546 + 0.546i)11-s + (0.0894 + 0.0894i)12-s − 1.26·13-s − 0.144i·15-s − 1.17·16-s + 0.368·18-s − 1.76i·19-s + (−0.0389 − 0.0389i)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: $\chi_{867} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.769 - 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.244009 + 0.676502i\)
\(L(\frac12)\) \(\approx\) \(0.244009 + 0.676502i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56iT - 2T^{2} \)
5 \( 1 + (-0.397 - 0.397i)T + 5iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + (1.81 - 1.81i)T - 11iT^{2} \)
13 \( 1 + 4.56T + 13T^{2} \)
19 \( 1 + 7.68iT - 19T^{2} \)
23 \( 1 + (4.63 - 4.63i)T - 23iT^{2} \)
29 \( 1 + (5.83 + 5.83i)T + 29iT^{2} \)
31 \( 1 + (3.62 + 3.62i)T + 31iT^{2} \)
37 \( 1 + (-2.20 - 2.20i)T + 37iT^{2} \)
41 \( 1 + (0.397 - 0.397i)T - 41iT^{2} \)
43 \( 1 + 7.68iT - 43T^{2} \)
47 \( 1 - 2.87T + 47T^{2} \)
53 \( 1 - 4.24iT - 53T^{2} \)
59 \( 1 + 1.12iT - 59T^{2} \)
61 \( 1 + (0.620 - 0.620i)T - 61iT^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (-7.24 - 7.24i)T + 71iT^{2} \)
73 \( 1 + (3.00 + 3.00i)T + 73iT^{2} \)
79 \( 1 + (-10.8 + 10.8i)T - 79iT^{2} \)
83 \( 1 - 9.12iT - 83T^{2} \)
89 \( 1 + 7.12T + 89T^{2} \)
97 \( 1 + (-7.86 - 7.86i)T + 97iT^{2} \)
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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

−9.838471346994297739051191528273, −9.255192598035686238996595455661, −7.70184983466921714979273886105, −7.19494837588564842897339972257, −6.17851492725465355796047266548, −5.05846373661442937479676915845, −4.06680287254957540196430691011, −2.60380660788013128392511111312, −2.07668325272879071148454634899, −0.32908862201683423479653519120, 2.06158094163340415550644358178, 3.55405977236080889960811369344, 4.91312903744356277656152536868, 5.51758853019361313292462455767, 6.24204223524156709736716905379, 7.27988571010311315346336550641, 7.941747059382026327394205988870, 8.824688615522628646958661755410, 9.806706970575036398541427196835, 10.57153629198441179209493555744

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