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Properties

Label 1-1148-1148.219-r0-0-0
Degree $1$
Conductor $1148$
Sign $0.797 - 0.602i$
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  + (0.258 + 0.965i)3-s + (−0.866 + 0.5i)5-s + (−0.866 + 0.5i)9-s + (−0.258 − 0.965i)11-s + (−0.707 + 0.707i)13-s + (−0.707 − 0.707i)15-s + (0.965 − 0.258i)17-s + (0.258 − 0.965i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)29-s + (−0.5 + 0.866i)31-s + (0.866 − 0.5i)33-s + (−0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)3-s + (−0.866 + 0.5i)5-s + (−0.866 + 0.5i)9-s + (−0.258 − 0.965i)11-s + (−0.707 + 0.707i)13-s + (−0.707 − 0.707i)15-s + (0.965 − 0.258i)17-s + (0.258 − 0.965i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)29-s + (−0.5 + 0.866i)31-s + (0.866 − 0.5i)33-s + (−0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.797 - 0.602i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 0.797 - 0.602i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7274163318 - 0.2438464819i\)
\(L(\frac12)\) \(\approx\) \(0.7274163318 - 0.2438464819i\)
\(L(1)\) \(\approx\) \(0.8090240076 + 0.1868119660i\)
\(L(1)\) \(\approx\) \(0.8090240076 + 0.1868119660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (0.965 - 0.258i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.258 - 0.965i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.965 + 0.258i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (0.965 + 0.258i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−21.03516197676437289442220565917, −20.45198672399286936071954563876, −19.80377049995886300165429638322, −19.11810815159522796170770274555, −18.43364230683042285725791516960, −17.51730433086261858528510118418, −16.91011619881771315575987606614, −15.924469162347539740088430455176, −14.94823556073249716712792866286, −14.56989412943466029769046598245, −13.29925586123937585605271938792, −12.739615985897094566543011423496, −12.035584456720816143202740698085, −11.537710250363049241793838113828, −10.16035128088154981023897839150, −9.45039868273556432147980110364, −8.24931012547986379159403316089, −7.688722698284024336988554611317, −7.30191177338297706523842382352, −5.947169871455773969196013304126, −5.216187604793508301577590694619, −4.024602375221827280602602397661, −3.170193120388507970027991401317, −2.03804958660968195296264019130, −1.07614569087597034667910425600, 0.33773002448313240569757345383, 2.28347768150698406446986357310, 3.21096573077590797510284064255, 3.8065620005518248969660516908, 4.82660841259753763571499729990, 5.569740172142670121998995257202, 6.84606227784919564887754843639, 7.62091749218917255963489117570, 8.58133910016915937224261152634, 9.19546282055516404181209730107, 10.297418526879450329886483066766, 10.855504568910040284676083264773, 11.64129231646855923642814146645, 12.37355189208537875446645201740, 13.7366299155215702482995062624, 14.37237025965927311053044891423, 14.94560385271065205915586873550, 15.967513477517116994393967517340, 16.27898369367578920062748398191, 17.11658929534796598355717625680, 18.34297263312919452419616907073, 19.01937465871361473815353054539, 19.72985770737428746206366404790, 20.391761680461675539540564256865, 21.35041151041143123397355142487

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