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Properties

Label 1-867-867.32-r1-0-0
Degree $1$
Conductor $867$
Sign $0.716 - 0.697i$
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  + (0.798 + 0.602i)2-s + (0.273 + 0.961i)4-s + (−0.403 + 0.914i)5-s + (−0.638 + 0.769i)7-s + (−0.361 + 0.932i)8-s + (−0.873 + 0.486i)10-s + (0.486 − 0.873i)11-s + (−0.932 − 0.361i)13-s + (−0.973 + 0.228i)14-s + (−0.850 + 0.526i)16-s + (0.798 − 0.602i)19-s + (−0.990 − 0.138i)20-s + (0.914 − 0.403i)22-s + (−0.638 + 0.769i)23-s + (−0.673 − 0.739i)25-s + (−0.526 − 0.850i)26-s + ⋯
L(s)  = 1  + (0.798 + 0.602i)2-s + (0.273 + 0.961i)4-s + (−0.403 + 0.914i)5-s + (−0.638 + 0.769i)7-s + (−0.361 + 0.932i)8-s + (−0.873 + 0.486i)10-s + (0.486 − 0.873i)11-s + (−0.932 − 0.361i)13-s + (−0.973 + 0.228i)14-s + (−0.850 + 0.526i)16-s + (0.798 − 0.602i)19-s + (−0.990 − 0.138i)20-s + (0.914 − 0.403i)22-s + (−0.638 + 0.769i)23-s + (−0.673 − 0.739i)25-s + (−0.526 − 0.850i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(93.1720\)
Root analytic conductor: \(93.1720\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 867,\ (1:\ ),\ 0.716 - 0.697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4430782645 - 0.1800120618i\)
\(L(\frac12)\) \(\approx\) \(0.4430782645 - 0.1800120618i\)
\(L(1)\) \(\approx\) \(0.9621595066 + 0.6408122444i\)
\(L(1)\) \(\approx\) \(0.9621595066 + 0.6408122444i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.798 + 0.602i)T \)
5 \( 1 + (-0.403 + 0.914i)T \)
7 \( 1 + (-0.638 + 0.769i)T \)
11 \( 1 + (0.486 - 0.873i)T \)
13 \( 1 + (-0.932 - 0.361i)T \)
19 \( 1 + (0.798 - 0.602i)T \)
23 \( 1 + (-0.638 + 0.769i)T \)
29 \( 1 + (0.873 + 0.486i)T \)
31 \( 1 + (-0.914 + 0.403i)T \)
37 \( 1 + (-0.948 - 0.317i)T \)
41 \( 1 + (-0.0461 - 0.998i)T \)
43 \( 1 + (-0.526 + 0.850i)T \)
47 \( 1 + (0.0922 - 0.995i)T \)
53 \( 1 + (0.995 - 0.0922i)T \)
59 \( 1 + (0.183 - 0.982i)T \)
61 \( 1 + (-0.824 - 0.565i)T \)
67 \( 1 + (-0.602 - 0.798i)T \)
71 \( 1 + (0.769 + 0.638i)T \)
73 \( 1 + (-0.228 + 0.973i)T \)
79 \( 1 + (-0.138 + 0.990i)T \)
83 \( 1 + (-0.673 - 0.739i)T \)
89 \( 1 + (0.932 - 0.361i)T \)
97 \( 1 + (0.769 + 0.638i)T \)
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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

−22.113517613849056548068608077277, −20.969796964330040326188892094426, −20.3256503788728604913137137406, −19.799588910059732120622999136162, −19.226648835405310964999449816476, −18.06042951812639408011983042015, −16.900148906881154733517047193318, −16.35142790350940883578822847849, −15.43497303593462753100885422155, −14.54237876851880938675011798643, −13.76223359512114059233394299825, −12.94305070136436551868823009102, −12.10990031571852157278663169517, −11.84903453296317161816951236981, −10.41955595998069815192453086461, −9.83078772311927672494305674656, −9.06008325280541801579229580780, −7.64363311716791768837158725581, −6.88440691264389874917330902832, −5.83490461131848565496528314504, −4.69750367736960610594395510178, −4.24461533409580671237363644539, −3.30077321651952855055382610372, −2.0257150041308160490848957367, −1.02508805110419167877935290514, 0.08232930066187390416586005823, 2.227455223767387413879677617277, 3.190548295200601755210400278, 3.62285127051649704769545053158, 5.062870245701151499919977277426, 5.7975820013584687651082216253, 6.71992531122933275833115056708, 7.32134004874934136260902759704, 8.34652442031870925504978382666, 9.25689067807409670536584089717, 10.397487030109410283388233503019, 11.528435269880301213641664094805, 11.98703632876007146941095864832, 12.89114844618229426063013958467, 13.95538538344043605248957114957, 14.42551533470019700088099487695, 15.47340219505200818104267409297, 15.78706909643720159931173028490, 16.74078228500206642391567003551, 17.74911899816417789303961741437, 18.45954454844531509308219202952, 19.551969754805806530232283375346, 19.96914906810008547966116465691, 21.646821731692621364798252148964, 21.747106955648484063920972021525

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