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Properties

Label 1-1148-1148.31-r0-0-0
Degree $1$
Conductor $1148$
Sign $-0.974 + 0.223i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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  + (0.5 + 0.866i)3-s + (0.104 + 0.994i)5-s + (−0.5 + 0.866i)9-s + (−0.104 + 0.994i)11-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.104 + 0.994i)17-s + (−0.669 − 0.743i)19-s + (0.978 + 0.207i)23-s + (−0.978 + 0.207i)25-s − 27-s + (0.809 − 0.587i)29-s + (−0.104 + 0.994i)31-s + (−0.913 + 0.406i)33-s + (−0.104 − 0.994i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.104 + 0.994i)5-s + (−0.5 + 0.866i)9-s + (−0.104 + 0.994i)11-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.104 + 0.994i)17-s + (−0.669 − 0.743i)19-s + (0.978 + 0.207i)23-s + (−0.978 + 0.207i)25-s − 27-s + (0.809 − 0.587i)29-s + (−0.104 + 0.994i)31-s + (−0.913 + 0.406i)33-s + (−0.104 − 0.994i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.974 + 0.223i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ -0.974 + 0.223i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1777597743 + 1.571297023i\)
\(L(\frac12)\) \(\approx\) \(0.1777597743 + 1.571297023i\)
\(L(1)\) \(\approx\) \(0.9228727028 + 0.7708811002i\)
\(L(1)\) \(\approx\) \(0.9228727028 + 0.7708811002i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.104 + 0.994i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (0.978 + 0.207i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (0.913 + 0.406i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−20.66657616759821791005417412389, −20.309315640334915927961940780353, −19.35694450482511635050411182444, −18.70518858074697639587223408288, −17.95361618300585824703259142110, −17.08798023658038664190648646768, −16.40631712381055665321498048099, −15.496479692287622288333481843, −14.60416194895819943507600364649, −13.62304407455992254954394689137, −13.228224202549225037569425624030, −12.45395165636756095508854289123, −11.70858116242679504305361506190, −10.728499320766888033807749651055, −9.59218626114447770026029845216, −8.71300584833967586550723198108, −8.26902740391063998358371524954, −7.44156106118575147097093893157, −6.30080801169787534152973377339, −5.63496816732451163094739487700, −4.61775453172753060883647820551, −3.38401995435109676361491511123, −2.6328098118314960539761795605, −1.3669840691390979638393015855, −0.60975074091431541122665254088, 1.865880832108328796518307658689, 2.558057644778022453954229539831, 3.623324035656934903611772622646, 4.31785931497290643887000870648, 5.25212156685537864083423886773, 6.478646104060145256004518066733, 7.09913074314470636782180121256, 8.183694706743098991915353132763, 9.05977268614123096996052049190, 9.768318535585433327635954989412, 10.6869018353011558639198699303, 11.02590321545579963121676886311, 12.193480770324579562763943558515, 13.25378907618395892427277944120, 14.09108772641153645615752042281, 14.72731912267197155200442582915, 15.36690975910697438988586793111, 15.96492957617568175299649274491, 17.1972288333995614214392813601, 17.577275124062726741849428107103, 18.841189050955416425806076783973, 19.31602636961488050370939597716, 20.097372670124941866185137393623, 21.12468154712205457119303099461, 21.54966908223127172907358447226

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