Properties

Label 1-1148-1148.327-r0-0-0
Degree $1$
Conductor $1148$
Sign $0.605 + 0.795i$
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 + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 13-s + 15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 13-s + 15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.538079717 + 0.7624680365i\)
\(L(\frac12)\) \(\approx\) \(1.538079717 + 0.7624680365i\)
\(L(1)\) \(\approx\) \(1.272844167 + 0.08827015491i\)
\(L(1)\) \(\approx\) \(1.272844167 + 0.08827015491i\)

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 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−21.08944930441852046345594614970, −20.40079230282130746342757296711, −20.09645384162736883692991797461, −18.80294815555291664736462461723, −18.231631244306730471197719359733, −16.981609229234316878182828086216, −16.54661191214029992130639383908, −15.746044882935623892616108737783, −15.20412125897820017071595346902, −13.872160671387444000364022104928, −13.60984948038648243688900642716, −12.83256329636133300837997192747, −11.45897555408821010499808532219, −10.94792718500418358928129812046, −9.94606313266839916028844639346, −9.10986851914178723671097920510, −8.68085259380252604327888862618, −7.84685903068439254395477473045, −6.5317181539228640477926404638, −5.43786650200604149252701489624, −4.96179966861733440862807213132, −3.92621765341618320939427380776, −3.003764913561583665675137761292, −2.0535818975590507988566872607, −0.63262202838960102264971423455, 1.505690533216996031706094673547, 2.023658838595023675338400886611, 3.17562464748980552560272518699, 3.83557843285783484441750594970, 5.45054308753465703670618833057, 6.128586641249302114443092405048, 7.06482246008139492243736290549, 7.58274848751537989027653170989, 8.60491032052114770612973781915, 9.429096064524144461759739917921, 10.37628026319877772391044285165, 11.12558237936639837787670629195, 12.10277605868317863230212834225, 13.05496983957448226906385391163, 13.46693893128170748569782556283, 14.4277882933334702357303429835, 14.97600100841294603803745976504, 15.79057137088339903602597241472, 17.037860204492732089658757064975, 17.934942022736359130538292986873, 18.18098803455023703922486127643, 19.03971003666964556144325451483, 19.74283090711064134537908225341, 20.69338930339328367067297933476, 21.20452182463848047803919823093

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