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Properties

Label 2-867-17.8-c1-0-0
Degree $2$
Conductor $867$
Sign $0.0881 + 0.996i$
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  + (−1.10 + 1.10i)2-s + (−0.382 + 0.923i)3-s − 0.438i·4-s + (−0.518 − 0.214i)5-s + (−0.597 − 1.44i)6-s + (−1.72 − 1.72i)8-s + (−0.707 − 0.707i)9-s + (0.810 − 0.335i)10-s + (0.980 + 2.36i)11-s + (0.405 + 0.167i)12-s + 4.56i·13-s + (0.397 − 0.397i)15-s + 4.68·16-s + 1.56·18-s + (−5.43 + 5.43i)19-s + (−0.0942 + 0.227i)20-s + ⋯
L(s)  = 1  + (−0.780 + 0.780i)2-s + (−0.220 + 0.533i)3-s − 0.219i·4-s + (−0.232 − 0.0961i)5-s + (−0.243 − 0.588i)6-s + (−0.609 − 0.609i)8-s + (−0.235 − 0.235i)9-s + (0.256 − 0.106i)10-s + (0.295 + 0.713i)11-s + (0.116 + 0.0484i)12-s + 1.26i·13-s + (0.102 − 0.102i)15-s + 1.17·16-s + 0.368·18-s + (−1.24 + 1.24i)19-s + (−0.0210 + 0.0508i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0881 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0881 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.0881 + 0.996i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (688, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.0881 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0646121 - 0.0591449i\)
\(L(\frac12)\) \(\approx\) \(0.0646121 - 0.0591449i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 \)
good2 \( 1 + (1.10 - 1.10i)T - 2iT^{2} \)
5 \( 1 + (0.518 + 0.214i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.980 - 2.36i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 4.56iT - 13T^{2} \)
19 \( 1 + (5.43 - 5.43i)T - 19iT^{2} \)
23 \( 1 + (2.51 + 6.06i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (7.61 + 3.15i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.96 + 4.73i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.19 + 2.88i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.518 + 0.214i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-5.43 - 5.43i)T + 43iT^{2} \)
47 \( 1 + 2.87iT - 47T^{2} \)
53 \( 1 + (-3.00 + 3.00i)T - 53iT^{2} \)
59 \( 1 + (0.794 + 0.794i)T + 59iT^{2} \)
61 \( 1 + (0.810 - 0.335i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-3.92 + 9.46i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-3.92 - 1.62i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (5.88 + 14.1i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (6.45 - 6.45i)T - 83iT^{2} \)
89 \( 1 + 7.12iT - 89T^{2} \)
97 \( 1 + (-10.2 - 4.25i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.50143053605812022645035857013, −9.705529104231204655961659422151, −9.135153556145115932701116697830, −8.235904029379028484762471869700, −7.57423843573689610816383116020, −6.45944931029736316481570974658, −6.02226619540642491124026681455, −4.35657143448548228596154062899, −3.94919355079944416592345080385, −2.11487893233679428099390806633, 0.05789090820620661383491020123, 1.35988289109890830483749543697, 2.60970332866445279366284480184, 3.63880186012587314252110014967, 5.29862350261364106127336922048, 5.95406165710193462021243342426, 7.10904202319673496469074032823, 8.043522480511448068683920971435, 8.752769590419821393454784368097, 9.553301721859826996887721841612

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