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Properties

Label 2-867-17.16-c1-0-16
Degree $2$
Conductor $867$
Sign $0.685 - 0.727i$
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  − 0.222·2-s + i·3-s − 1.95·4-s − 0.636i·5-s − 0.222i·6-s + 1.72i·7-s + 0.877·8-s − 9-s + 0.141i·10-s − 4.95i·11-s − 1.95i·12-s + 2.50·13-s − 0.384i·14-s + 0.636·15-s + 3.70·16-s + ⋯
L(s)  = 1  − 0.157·2-s + 0.577i·3-s − 0.975·4-s − 0.284i·5-s − 0.0907i·6-s + 0.653i·7-s + 0.310·8-s − 0.333·9-s + 0.0447i·10-s − 1.49i·11-s − 0.563i·12-s + 0.695·13-s − 0.102i·14-s + 0.164·15-s + 0.926·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.685 - 0.727i$
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: \((2,\ 867,\ (\ :1/2),\ 0.685 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02422 + 0.442016i\)
\(L(\frac12)\) \(\approx\) \(1.02422 + 0.442016i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.222T + 2T^{2} \)
5 \( 1 + 0.636iT - 5T^{2} \)
7 \( 1 - 1.72iT - 7T^{2} \)
11 \( 1 + 4.95iT - 11T^{2} \)
13 \( 1 - 2.50T + 13T^{2} \)
19 \( 1 - 0.950T + 19T^{2} \)
23 \( 1 - 2.12iT - 23T^{2} \)
29 \( 1 - 9.58iT - 29T^{2} \)
31 \( 1 - 5.27iT - 31T^{2} \)
37 \( 1 - 8.48iT - 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 7.15T + 43T^{2} \)
47 \( 1 - 8.10T + 47T^{2} \)
53 \( 1 - 6.44T + 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 + 2.96iT - 61T^{2} \)
67 \( 1 - 7.70T + 67T^{2} \)
71 \( 1 + 0.384iT - 71T^{2} \)
73 \( 1 - 6.51iT - 73T^{2} \)
79 \( 1 - 2.37iT - 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 1.98T + 89T^{2} \)
97 \( 1 - 3.04iT - 97T^{2} \)
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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

−10.27496446525046136201405542784, −9.131121883394248122240762899129, −8.726205967856213014149595823774, −8.252508117464996070866404648250, −6.75377667898946242936396216733, −5.48535618120529039241249069822, −5.19063758572564262818027138862, −3.84025121433295768013224260462, −3.10747731112498036310573454649, −1.05068469116769555930630762639, 0.791366050677559724548620545942, 2.27307093744709299731785477270, 3.83706373683543313800647803575, 4.54265557974530718873914560142, 5.69646911380748110972694175040, 6.79047751807200797398600474612, 7.54069505466690126324911556782, 8.274060904828318324936762652132, 9.266347400143926297920358329598, 9.986332147858272636010092037781

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