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Properties

Label 2-867-51.2-c0-0-1
Degree $2$
Conductor $867$
Sign $0.999 - 0.0340i$
Analytic cond. $0.432689$
Root an. cond. $0.657791$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)3-s + i·4-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (0.382 + 0.923i)12-s + i·13-s − 16-s + (−0.707 − 0.707i)19-s − i·21-s + (−0.707 + 0.707i)25-s + (0.382 − 0.923i)27-s + (0.923 + 0.382i)28-s + (0.923 − 0.382i)31-s + (0.707 + 0.707i)36-s + (−0.923 + 0.382i)37-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + i·4-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (0.382 + 0.923i)12-s + i·13-s − 16-s + (−0.707 − 0.707i)19-s − i·21-s + (−0.707 + 0.707i)25-s + (0.382 − 0.923i)27-s + (0.923 + 0.382i)28-s + (0.923 − 0.382i)31-s + (0.707 + 0.707i)36-s + (−0.923 + 0.382i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.999 - 0.0340i$
Analytic conductor: \(0.432689\)
Root analytic conductor: \(0.657791\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :0),\ 0.999 - 0.0340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.349514988\)
\(L(\frac12)\) \(\approx\) \(1.349514988\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 \)
good2 \( 1 - iT^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.382 + 0.923i)T + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 - iT - T^{2} \)
19 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (-0.923 + 0.382i)T + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (0.923 - 0.382i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.382 - 0.923i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.765 + 1.84i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (1.84 + 0.765i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.382 + 0.923i)T + (-0.707 + 0.707i)T^{2} \)
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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

−10.27213161339090863085986148469, −9.273900036484333483502586743378, −8.605601929555690190476243621084, −7.80938404341013340971277432423, −7.16161623714469449351575662172, −6.47107483049690065272518711718, −4.58907416213623796132188266792, −3.98644323356187614159813774816, −2.94872538681356975298446018471, −1.75457076419915184541708130788, 1.77081069793684303000424648264, 2.71863640754289543396787989725, 4.05662387403405905650983266443, 5.13058773567342667681456111058, 5.80551743226326293318391928681, 6.93522262500807509653222387288, 8.246226343347726028618316167014, 8.542004730646228339499554545335, 9.641061789952397627910079882701, 10.19857062674171133571221915073

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