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Properties

Label 1-2e10-1024.125-r0-0-0
Degree $1$
Conductor $1024$
Sign $0.237 - 0.971i$
Analytic cond. $4.75543$
Root an. cond. $4.75543$
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  + (0.844 − 0.534i)3-s + (0.997 + 0.0735i)5-s + (−0.671 + 0.740i)7-s + (0.427 − 0.903i)9-s + (−0.999 + 0.0245i)11-s + (0.313 − 0.949i)13-s + (0.881 − 0.471i)15-s + (0.471 − 0.881i)17-s + (0.122 − 0.992i)19-s + (−0.170 + 0.985i)21-s + (−0.514 − 0.857i)23-s + (0.989 + 0.146i)25-s + (−0.122 − 0.992i)27-s + (−0.932 + 0.359i)29-s + (−0.831 − 0.555i)31-s + ⋯
L(s)  = 1  + (0.844 − 0.534i)3-s + (0.997 + 0.0735i)5-s + (−0.671 + 0.740i)7-s + (0.427 − 0.903i)9-s + (−0.999 + 0.0245i)11-s + (0.313 − 0.949i)13-s + (0.881 − 0.471i)15-s + (0.471 − 0.881i)17-s + (0.122 − 0.992i)19-s + (−0.170 + 0.985i)21-s + (−0.514 − 0.857i)23-s + (0.989 + 0.146i)25-s + (−0.122 − 0.992i)27-s + (−0.932 + 0.359i)29-s + (−0.831 − 0.555i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.237 - 0.971i$
Analytic conductor: \(4.75543\)
Root analytic conductor: \(4.75543\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1024,\ (0:\ ),\ 0.237 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.622446030 - 1.274197698i\)
\(L(\frac12)\) \(\approx\) \(1.622446030 - 1.274197698i\)
\(L(1)\) \(\approx\) \(1.402939552 - 0.4260179463i\)
\(L(1)\) \(\approx\) \(1.402939552 - 0.4260179463i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.844 - 0.534i)T \)
5 \( 1 + (0.997 + 0.0735i)T \)
7 \( 1 + (-0.671 + 0.740i)T \)
11 \( 1 + (-0.999 + 0.0245i)T \)
13 \( 1 + (0.313 - 0.949i)T \)
17 \( 1 + (0.471 - 0.881i)T \)
19 \( 1 + (0.122 - 0.992i)T \)
23 \( 1 + (-0.514 - 0.857i)T \)
29 \( 1 + (-0.932 + 0.359i)T \)
31 \( 1 + (-0.831 - 0.555i)T \)
37 \( 1 + (-0.266 + 0.963i)T \)
41 \( 1 + (0.989 - 0.146i)T \)
43 \( 1 + (-0.219 - 0.975i)T \)
47 \( 1 + (0.995 + 0.0980i)T \)
53 \( 1 + (0.932 + 0.359i)T \)
59 \( 1 + (-0.949 + 0.313i)T \)
61 \( 1 + (0.985 - 0.170i)T \)
67 \( 1 + (0.575 - 0.817i)T \)
71 \( 1 + (0.941 + 0.336i)T \)
73 \( 1 + (0.671 + 0.740i)T \)
79 \( 1 + (0.773 + 0.634i)T \)
83 \( 1 + (-0.266 - 0.963i)T \)
89 \( 1 + (-0.514 + 0.857i)T \)
97 \( 1 + (0.195 + 0.980i)T \)
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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

−21.39489911038140625493647526687, −21.14816850997801321277221173959, −20.32524891333684945306446649798, −19.48423110686290371049962897129, −18.76598033775385475471382373177, −17.928847859987791569650201541417, −16.7422323425119560626994326577, −16.41100223675016513428989636487, −15.51570067674504965275050292412, −14.40638017798146316644882580124, −13.980451223156855509658759002782, −13.16027625673214151275442932355, −12.62011598286276036654515903545, −11.0223789179131353148857603753, −10.32872155562091354940811817339, −9.71609570544482118413110623940, −9.06537889729188487906181238730, −7.99206467967792566687364670658, −7.249587650433975022481066273398, −6.061338507373228481300478050895, −5.32828183275592570753344040600, −4.05390518659985622184402055789, −3.48628254408446973593836610196, −2.29924808088627459774463991801, −1.51597913401328277744475011772, 0.75714396230694375538106664408, 2.28517694541316770813435171598, 2.61554353503612458541876186553, 3.547954744958715936109275132015, 5.164172860224389618754184172327, 5.79631872417755850485706032023, 6.765579376938645721460738900070, 7.60181957861303384782511374413, 8.58487566434518760964405147050, 9.301112961422473680628180720802, 9.96494255124376987619707807373, 10.88080715761270826327654674996, 12.23980265042292691621462323826, 12.89173886978591861522889004198, 13.421229882953352807625324760564, 14.17257338956634245839572544056, 15.181135382478172136052624863457, 15.68956452615651360389942295944, 16.750203531274034956284044283655, 17.877843973357384169676640716262, 18.45675102378219958460703314706, 18.76813824021911424394410370147, 20.11712762651035506172180851339, 20.48085006402930890084010251816, 21.34415838374144392533938145160

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