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Properties

Label 1-43-43.9-r0-0-0
Degree $1$
Conductor $43$
Sign $0.0689 + 0.997i$
Analytic cond. $0.199691$
Root an. cond. $0.199691$
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  + (−0.222 + 0.974i)2-s + (0.955 + 0.294i)3-s + (−0.900 − 0.433i)4-s + (0.365 + 0.930i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + (0.623 − 0.781i)8-s + (0.826 + 0.563i)9-s + (−0.988 + 0.149i)10-s + (−0.900 + 0.433i)11-s + (−0.733 − 0.680i)12-s + (−0.988 − 0.149i)13-s + (0.955 − 0.294i)14-s + (0.0747 + 0.997i)15-s + (0.623 + 0.781i)16-s + (0.365 − 0.930i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (0.955 + 0.294i)3-s + (−0.900 − 0.433i)4-s + (0.365 + 0.930i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + (0.623 − 0.781i)8-s + (0.826 + 0.563i)9-s + (−0.988 + 0.149i)10-s + (−0.900 + 0.433i)11-s + (−0.733 − 0.680i)12-s + (−0.988 − 0.149i)13-s + (0.955 − 0.294i)14-s + (0.0747 + 0.997i)15-s + (0.623 + 0.781i)16-s + (0.365 − 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0689 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0689 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.0689 + 0.997i$
Analytic conductor: \(0.199691\)
Root analytic conductor: \(0.199691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (0:\ ),\ 0.0689 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6494733595 + 0.6061340416i\)
\(L(\frac12)\) \(\approx\) \(0.6494733595 + 0.6061340416i\)
\(L(1)\) \(\approx\) \(0.8840804779 + 0.5570339266i\)
\(L(1)\) \(\approx\) \(0.8840804779 + 0.5570339266i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (0.955 + 0.294i)T \)
5 \( 1 + (0.365 + 0.930i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
17 \( 1 + (0.365 - 0.930i)T \)
19 \( 1 + (0.826 - 0.563i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (0.955 - 0.294i)T \)
31 \( 1 + (-0.733 - 0.680i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (0.0747 + 0.997i)T \)
73 \( 1 + (-0.988 - 0.149i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.955 + 0.294i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−34.83082789509458443117185264804, −32.566591500663159248576126307793, −31.72250825152723560211843516363, −31.07008018141388643286914471576, −29.425542948752854178789956300070, −28.81062281480994805516433735998, −27.41872210335725215569764291478, −26.13352204946004327089705011453, −25.06635367429451204434867378846, −23.76076005685048328672399913772, −21.782143920515970964666377610880, −21.069918258973434878781324514726, −19.81431887888280111434649104850, −18.96576016876800751162594305616, −17.69713717739764711820591696888, −15.98840629311402676778799692859, −14.18268969754224497370229540739, −12.90853069925690055644963294595, −12.23551693559762602526863768606, −10.00855768219029390414381628208, −9.04407987196600115838848901355, −7.94832510952212288593065451181, −5.27880869697051129252511302849, −3.29713646278602387968652393280, −1.86260340148840904965211612844, 2.93728946461447657819054500970, 4.80975307769848090140862527223, 6.917270166114516528455614197966, 7.73518149505279361901558572891, 9.63940247179466660017246297637, 10.281008694734744769443212737, 13.22718222550756434011132963520, 14.154395654900171166207821086032, 15.17140128180055920415883080945, 16.372180496884025786348018572372, 17.91874482058518616747235330299, 19.02274130929544205458238444316, 20.31562046889861429122162685200, 22.0201033628661463292317480403, 23.05184130776454648038264636272, 24.55993959409408064107712036163, 25.73054907281858974199303257266, 26.43444922253288544490396421459, 27.16382049165488375404437231824, 29.08159497846115804865423718746, 30.5337826387271693403400679518, 31.669960729134348572113021300050, 32.78123595018292607149697730052, 33.60749162493852179142749689309, 34.743177269842302173390371484376

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