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Properties

Label 1-43-43.34-r1-0-0
Degree $1$
Conductor $43$
Sign $-0.351 + 0.936i$
Analytic cond. $4.62099$
Root an. cond. $4.62099$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $-0.351 + 0.936i$
Analytic conductor: \(4.62099\)
Root analytic conductor: \(4.62099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (1:\ ),\ -0.351 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1250375501 - 0.1805530239i\)
\(L(\frac12)\) \(\approx\) \(-0.1250375501 - 0.1805530239i\)
\(L(1)\) \(\approx\) \(0.4275115564 - 0.3748081313i\)
\(L(1)\) \(\approx\) \(0.4275115564 - 0.3748081313i\)

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

−34.49681834576204887027177984933, −34.21193183255826171626548445883, −33.17471762289274137055758676357, −31.94360930159370394327566410156, −30.5167949925419022818496158567, −29.46190713043536766419962681439, −27.65745932255504095658480468503, −26.835276573044053013835297554743, −25.96085633523626948889151544693, −23.85034845937961424691500453278, −23.607797341067793153549119689573, −22.255206404919201343271479106734, −21.344753184180159899802725286912, −19.04575664179352193173705494481, −17.73192081559039749101038780988, −16.88964009332190311468665058748, −15.53461791301013505519421119880, −14.52793885838560990517512245428, −12.95054336231972744487933334669, −11.24293473418147683348526763184, −10.06599665881085891590828952233, −7.80648260500463909647171327985, −6.76247294655105446528772989248, −5.25453097316806927785063194337, −3.8383850762264910680237064742, 0.13836761956586586279494808628, 2.078374050380864804405822345808, 4.66756977274747501083708973541, 5.46224488549848915982855278291, 7.95562295066321424785749238056, 9.631923844781590780236393415990, 11.16179887148027960824569002792, 12.29861523062645019431849770079, 12.872818698552231818394818570267, 14.91984141500100053197911641867, 16.58175704348090475565194994067, 17.9663232599222608218886977102, 18.896087428665260191071388336226, 20.45053245936592994889796577843, 21.46751832391095958745176980367, 22.70766515956387291158466422098, 23.77679510664625103613556732050, 24.77300320420201144499952107375, 27.159863332550893205816978778871, 28.00405203363356451100227785227, 28.76474748285393266686628173529, 29.809458603941575022502593357712, 31.251989077111195308091297736, 31.94643472014477988468026335569, 33.57871806628068491194341301101

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