Properties

Label 1-968-968.389-r0-0-0
Degree $1$
Conductor $968$
Sign $-0.631 + 0.775i$
Analytic cond. $4.49537$
Root an. cond. $4.49537$
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  + (0.809 + 0.587i)3-s + (0.736 + 0.676i)5-s + (−0.998 + 0.0570i)7-s + (0.309 + 0.951i)9-s + (0.564 + 0.825i)13-s + (0.198 + 0.980i)15-s + (−0.921 + 0.389i)17-s + (−0.974 + 0.226i)19-s + (−0.841 − 0.540i)21-s + (0.841 − 0.540i)23-s + (0.0855 + 0.996i)25-s + (−0.309 + 0.951i)27-s + (−0.0855 + 0.996i)29-s + (−0.0285 − 0.999i)31-s + (−0.774 − 0.633i)35-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (0.736 + 0.676i)5-s + (−0.998 + 0.0570i)7-s + (0.309 + 0.951i)9-s + (0.564 + 0.825i)13-s + (0.198 + 0.980i)15-s + (−0.921 + 0.389i)17-s + (−0.974 + 0.226i)19-s + (−0.841 − 0.540i)21-s + (0.841 − 0.540i)23-s + (0.0855 + 0.996i)25-s + (−0.309 + 0.951i)27-s + (−0.0855 + 0.996i)29-s + (−0.0285 − 0.999i)31-s + (−0.774 − 0.633i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.631 + 0.775i$
Analytic conductor: \(4.49537\)
Root analytic conductor: \(4.49537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 968,\ (0:\ ),\ -0.631 + 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7408241588 + 1.557504119i\)
\(L(\frac12)\) \(\approx\) \(0.7408241588 + 1.557504119i\)
\(L(1)\) \(\approx\) \(1.153612588 + 0.6422643497i\)
\(L(1)\) \(\approx\) \(1.153612588 + 0.6422643497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.736 + 0.676i)T \)
7 \( 1 + (-0.998 + 0.0570i)T \)
13 \( 1 + (0.564 + 0.825i)T \)
17 \( 1 + (-0.921 + 0.389i)T \)
19 \( 1 + (-0.974 + 0.226i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (-0.0855 + 0.996i)T \)
31 \( 1 + (-0.0285 - 0.999i)T \)
37 \( 1 + (-0.941 + 0.336i)T \)
41 \( 1 + (-0.466 + 0.884i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.696 - 0.717i)T \)
53 \( 1 + (0.254 - 0.967i)T \)
59 \( 1 + (0.466 + 0.884i)T \)
61 \( 1 + (-0.897 - 0.441i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (0.516 + 0.856i)T \)
73 \( 1 + (-0.362 + 0.931i)T \)
79 \( 1 + (0.993 + 0.113i)T \)
83 \( 1 + (-0.774 + 0.633i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.736 + 0.676i)T \)
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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

−21.197035888471932793079710240453, −20.68325904681970954390916942090, −19.779259757600309307295739852178, −19.33836658291757819690082423419, −18.32643447356758073973694067479, −17.59248272958534635202936411833, −16.87901213656001333038774744327, −15.69292218639224361324756264286, −15.304253760835500960924275669318, −13.95975355603302731297355269714, −13.45315171288248152402205708420, −12.84751536479751500290330283568, −12.263511268851611273531372067472, −10.85349832249384959654213107647, −9.94904977564813308038805464145, −9.02894473872337282969340377891, −8.70760580551579958994429133074, −7.536068408373215789720969143612, −6.57614861578616989154698478275, −5.96750048543249216265364811626, −4.75603563209306535326092053305, −3.59441102763027547316031060094, −2.72887754756198405939225197566, −1.79819526320770123595172960817, −0.62275681955412326839235234600, 1.79330191599773871278444901641, 2.558201800891831480373228879805, 3.483204047151788108652542598940, 4.25329215993143006115928839831, 5.486591087358445234970836602985, 6.58210285762677355983330330357, 7.00406816603596311428143778731, 8.57347688791574198603080573954, 8.977879896655847844791606961334, 9.94394199865681301690003820115, 10.51529290124925877149453082629, 11.32648911930601934700418906985, 12.812011045870876179014829478553, 13.34970992746815515880386394748, 14.08246165009563796470884012314, 14.96337295508850255482055903014, 15.49304958718601181467597270778, 16.556786202305253422949034836348, 17.06031404238588035580774842268, 18.42702273035003894999975681401, 18.8908256537646898697078672106, 19.63826107093769353234685199069, 20.54302248526540729727970614598, 21.30866637922181861844548300875, 21.95376696277231794004072658627

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