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Properties

Label 2-867-17.16-c1-0-24
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  − 2.06·2-s − i·3-s + 2.27·4-s − 0.347i·5-s + 2.06i·6-s + 4.33i·7-s − 0.558·8-s − 9-s + 0.718i·10-s + 0.729i·11-s − 2.27i·12-s − 5.40·13-s − 8.96i·14-s − 0.347·15-s − 3.38·16-s + ⋯
L(s)  = 1  − 1.46·2-s − 0.577i·3-s + 1.13·4-s − 0.155i·5-s + 0.843i·6-s + 1.63i·7-s − 0.197·8-s − 0.333·9-s + 0.227i·10-s + 0.220i·11-s − 0.655i·12-s − 1.49·13-s − 2.39i·14-s − 0.0897·15-s − 0.846·16-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

\(L(1)\) \(\approx\) \(0.0985424 - 0.228340i\)
\(L(\frac12)\) \(\approx\) \(0.0985424 - 0.228340i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.06T + 2T^{2} \)
5 \( 1 + 0.347iT - 5T^{2} \)
7 \( 1 - 4.33iT - 7T^{2} \)
11 \( 1 - 0.729iT - 11T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
19 \( 1 + 3.27T + 19T^{2} \)
23 \( 1 + 3.55iT - 23T^{2} \)
29 \( 1 + 4.38iT - 29T^{2} \)
31 \( 1 + 3.30iT - 31T^{2} \)
37 \( 1 + 2.87iT - 37T^{2} \)
41 \( 1 + 11.8iT - 41T^{2} \)
43 \( 1 + 3.74T + 43T^{2} \)
47 \( 1 - 0.476T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + 5.29T + 59T^{2} \)
61 \( 1 + 3.54iT - 61T^{2} \)
67 \( 1 + 1.55T + 67T^{2} \)
71 \( 1 + 8.96iT - 71T^{2} \)
73 \( 1 + 4.09iT - 73T^{2} \)
79 \( 1 + 0.812iT - 79T^{2} \)
83 \( 1 + 7.21T + 83T^{2} \)
89 \( 1 + 9.77T + 89T^{2} \)
97 \( 1 + 1.77iT - 97T^{2} \)
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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

−9.612890305957733696565886350666, −8.902498420381374562414647872445, −8.404998612018054317349490405988, −7.49749152452854028415541178926, −6.72158873037056165935641277703, −5.67873468197744739865266051772, −4.63973036020868270686678481555, −2.55096462467407635405723565890, −2.04011698470897341464551252552, −0.20693115462134271557457953722, 1.24986158930048473544075986745, 2.87503973219968599193716910157, 4.19723647935635740454122131483, 5.01940310006609215810070147743, 6.70700838744006876417398073750, 7.22398871012393443038316163306, 8.034547398905639411562346048735, 8.889694975560901624570596835447, 9.852729291469090919974654058065, 10.20004791943777052815364768679

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