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Properties

Label 1-867-867.662-r1-0-0
Degree $1$
Conductor $867$
Sign $-0.493 + 0.869i$
Analytic cond. $93.1720$
Root an. cond. $93.1720$
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  + (0.850 − 0.526i)2-s + (0.445 − 0.895i)4-s + (0.0922 − 0.995i)5-s + (−0.932 + 0.361i)7-s + (−0.0922 − 0.995i)8-s + (−0.445 − 0.895i)10-s + (0.445 − 0.895i)11-s + (0.0922 + 0.995i)13-s + (−0.602 + 0.798i)14-s + (−0.602 − 0.798i)16-s + (−0.850 − 0.526i)19-s + (−0.850 − 0.526i)20-s + (−0.0922 − 0.995i)22-s + (0.932 − 0.361i)23-s + (−0.982 − 0.183i)25-s + (0.602 + 0.798i)26-s + ⋯
L(s)  = 1  + (0.850 − 0.526i)2-s + (0.445 − 0.895i)4-s + (0.0922 − 0.995i)5-s + (−0.932 + 0.361i)7-s + (−0.0922 − 0.995i)8-s + (−0.445 − 0.895i)10-s + (0.445 − 0.895i)11-s + (0.0922 + 0.995i)13-s + (−0.602 + 0.798i)14-s + (−0.602 − 0.798i)16-s + (−0.850 − 0.526i)19-s + (−0.850 − 0.526i)20-s + (−0.0922 − 0.995i)22-s + (0.932 − 0.361i)23-s + (−0.982 − 0.183i)25-s + (0.602 + 0.798i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.493 + 0.869i$
Analytic conductor: \(93.1720\)
Root analytic conductor: \(93.1720\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (662, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 867,\ (1:\ ),\ -0.493 + 0.869i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6028083051 - 1.035411913i\)
\(L(\frac12)\) \(\approx\) \(-0.6028083051 - 1.035411913i\)
\(L(1)\) \(\approx\) \(1.061247584 - 0.8284413757i\)
\(L(1)\) \(\approx\) \(1.061247584 - 0.8284413757i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.850 - 0.526i)T \)
5 \( 1 + (0.0922 - 0.995i)T \)
7 \( 1 + (-0.932 + 0.361i)T \)
11 \( 1 + (0.445 - 0.895i)T \)
13 \( 1 + (0.0922 + 0.995i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (0.932 - 0.361i)T \)
29 \( 1 + (0.445 - 0.895i)T \)
31 \( 1 + (-0.0922 - 0.995i)T \)
37 \( 1 + (0.273 + 0.961i)T \)
41 \( 1 + (-0.982 + 0.183i)T \)
43 \( 1 + (-0.602 + 0.798i)T \)
47 \( 1 + (-0.932 - 0.361i)T \)
53 \( 1 + (-0.932 + 0.361i)T \)
59 \( 1 + (-0.739 - 0.673i)T \)
61 \( 1 + (-0.739 + 0.673i)T \)
67 \( 1 + (-0.850 - 0.526i)T \)
71 \( 1 + (0.932 - 0.361i)T \)
73 \( 1 + (0.602 + 0.798i)T \)
79 \( 1 + (0.850 + 0.526i)T \)
83 \( 1 + (0.982 + 0.183i)T \)
89 \( 1 + (-0.0922 + 0.995i)T \)
97 \( 1 + (-0.932 + 0.361i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.49713184826363928787320306633, −21.893132954652731125989518607899, −20.97832387765909164267442966933, −20.008590395836281380265772475588, −19.39704646750713597731447918456, −18.21184592718962310435461237850, −17.468650877114404405143932810429, −16.71690793021953799903945743370, −15.72991351656046275554884148045, −15.075452649148571133877546098607, −14.45436271303284494615960373518, −13.56250262517221183694998448953, −12.74611972570163571696546236558, −12.16537791524011874120065030773, −10.8897716621784557073851481181, −10.34198197208051287153919715212, −9.22419227980199949016482794356, −8.011396480982059834903689882004, −7.04202014588216762296722650491, −6.65709304579166333985416503516, −5.72915510044530237541054119078, −4.65566655046427556472433971793, −3.48971371888940711824287046344, −3.09329556010653887190521742777, −1.82556158349008412156264572228, 0.18065143011909198658936372681, 1.25035460942522293960971148536, 2.37088644219821991812532318063, 3.38909727254607152222513826096, 4.32801660081705220488216420710, 5.084840761235885699240846374377, 6.29994710488265070774822665502, 6.51884859884034276731454203646, 8.242984948479143014570261488197, 9.21295435638959757980433744895, 9.68784123337073047754568893561, 10.95255472724302082289859089393, 11.72525625399629195389674873152, 12.41527006864017124203110748478, 13.36241420743322905781204112771, 13.59550217012223237004996517811, 14.871376748623295108534285010880, 15.595588280262303111881316669918, 16.57048313324638437978504558393, 16.895969604266440371190218149919, 18.53311677883504845192824401177, 19.22043251832403334590835772937, 19.70736686663009609115035332902, 20.68805581007513650214619381388, 21.44056284071707028653129242397

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