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Properties

Label 1-1063-1063.30-r0-0-0
Degree $1$
Conductor $1063$
Sign $-0.992 + 0.124i$
Analytic cond. $4.93655$
Root an. cond. $4.93655$
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  + (0.265 − 0.963i)2-s + (0.693 − 0.720i)3-s + (−0.858 − 0.512i)4-s + (0.00887 − 0.999i)5-s + (−0.510 − 0.860i)6-s + (0.766 − 0.642i)7-s + (−0.722 + 0.691i)8-s + (−0.0384 − 0.999i)9-s + (−0.961 − 0.274i)10-s + (0.945 + 0.325i)11-s + (−0.964 + 0.263i)12-s + (0.937 + 0.347i)13-s + (−0.415 − 0.909i)14-s + (−0.714 − 0.699i)15-s + (0.474 + 0.880i)16-s + (−0.0266 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.265 − 0.963i)2-s + (0.693 − 0.720i)3-s + (−0.858 − 0.512i)4-s + (0.00887 − 0.999i)5-s + (−0.510 − 0.860i)6-s + (0.766 − 0.642i)7-s + (−0.722 + 0.691i)8-s + (−0.0384 − 0.999i)9-s + (−0.961 − 0.274i)10-s + (0.945 + 0.325i)11-s + (−0.964 + 0.263i)12-s + (0.937 + 0.347i)13-s + (−0.415 − 0.909i)14-s + (−0.714 − 0.699i)15-s + (0.474 + 0.880i)16-s + (−0.0266 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1063\)
Sign: $-0.992 + 0.124i$
Analytic conductor: \(4.93655\)
Root analytic conductor: \(4.93655\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1063} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1063,\ (0:\ ),\ -0.992 + 0.124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1513464827 - 2.421524324i\)
\(L(\frac12)\) \(\approx\) \(-0.1513464827 - 2.421524324i\)
\(L(1)\) \(\approx\) \(0.8120758602 - 1.403259450i\)
\(L(1)\) \(\approx\) \(0.8120758602 - 1.403259450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1063 \( 1 \)
good2 \( 1 + (0.265 - 0.963i)T \)
3 \( 1 + (0.693 - 0.720i)T \)
5 \( 1 + (0.00887 - 0.999i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (0.945 + 0.325i)T \)
13 \( 1 + (0.937 + 0.347i)T \)
17 \( 1 + (-0.0266 - 0.999i)T \)
19 \( 1 + (-0.109 - 0.994i)T \)
23 \( 1 + (0.991 - 0.129i)T \)
29 \( 1 + (-0.819 + 0.572i)T \)
31 \( 1 + (0.277 + 0.960i)T \)
37 \( 1 + (0.631 + 0.775i)T \)
41 \( 1 + (0.989 + 0.141i)T \)
43 \( 1 + (0.959 - 0.280i)T \)
47 \( 1 + (-0.935 - 0.353i)T \)
53 \( 1 + (-0.976 - 0.217i)T \)
59 \( 1 + (-0.973 + 0.228i)T \)
61 \( 1 + (0.795 + 0.605i)T \)
67 \( 1 + (-0.898 - 0.439i)T \)
71 \( 1 + (0.484 + 0.874i)T \)
73 \( 1 + (-0.540 + 0.841i)T \)
79 \( 1 + (-0.144 + 0.989i)T \)
83 \( 1 + (0.750 + 0.660i)T \)
89 \( 1 + (-0.271 + 0.962i)T \)
97 \( 1 + (-0.978 - 0.205i)T \)
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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

−21.99853066160213968026410339505, −21.24570463267916761561457327903, −20.73265386387437181208406537508, −19.19812412711849138850833756793, −18.91074356418547860959566700448, −17.878539338387859992284372880904, −17.13980521079547892109513463256, −16.237917223435350702066293607658, −15.38080469910166207479537805246, −14.78876058809301654571046453632, −14.46131928642766165174782279441, −13.62289939591152062997319511519, −12.686633582475453333145381089509, −11.38311182275406394144528131316, −10.79984711122679357042253256053, −9.600707760845306126002645900075, −8.944449428601953586003975971760, −8.06073957710854989113962026464, −7.57754229191483618868178336091, −6.08208891862986770284400632425, −5.880841842726307816341961690542, −4.45841916049283088243617385462, −3.77875855997733972121171781485, −3.0208999094494044418657505376, −1.73728733273945151279413753765, 1.065310178853006626850118565207, 1.27167860888542317492402557524, 2.44935683590259627462652736505, 3.58532211984012021639243303676, 4.39096542571697653279771693207, 5.11501240425971115307573056840, 6.43876196672010050345167253952, 7.403344788722783873764086873123, 8.514892124434569693500311818617, 9.01446119860344528544125019114, 9.65425586430243708795298825397, 11.10634272515175693170484807821, 11.5501706835328013408918281431, 12.49933133309068112021620931688, 13.17350924120269914252292837709, 13.82987528226126057547449036345, 14.39960846610296383347639348057, 15.334882689001345788914070156068, 16.60167901349608430387420796368, 17.550488162388160591192037224666, 18.020628375774657711904462169522, 19.0327902344839993522448687866, 19.76602124522518754601797394490, 20.343721554095177258962884095472, 20.8214286480799773598988468237

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