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Properties

Label 1-43-43.10-r0-0-0
Degree $1$
Conductor $43$
Sign $0.118 + 0.992i$
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  + (−0.900 + 0.433i)2-s + (0.0747 + 0.997i)3-s + (0.623 − 0.781i)4-s + (0.955 − 0.294i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (−0.222 + 0.974i)8-s + (−0.988 + 0.149i)9-s + (−0.733 + 0.680i)10-s + (0.623 + 0.781i)11-s + (0.826 + 0.563i)12-s + (−0.733 − 0.680i)13-s + (0.0747 − 0.997i)14-s + (0.365 + 0.930i)15-s + (−0.222 − 0.974i)16-s + (0.955 + 0.294i)17-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.0747 + 0.997i)3-s + (0.623 − 0.781i)4-s + (0.955 − 0.294i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (−0.222 + 0.974i)8-s + (−0.988 + 0.149i)9-s + (−0.733 + 0.680i)10-s + (0.623 + 0.781i)11-s + (0.826 + 0.563i)12-s + (−0.733 − 0.680i)13-s + (0.0747 − 0.997i)14-s + (0.365 + 0.930i)15-s + (−0.222 − 0.974i)16-s + (0.955 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.118 + 0.992i$
Analytic conductor: \(0.199691\)
Root analytic conductor: \(0.199691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (0:\ ),\ 0.118 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4647456281 + 0.4125800977i\)
\(L(\frac12)\) \(\approx\) \(0.4647456281 + 0.4125800977i\)
\(L(1)\) \(\approx\) \(0.6603551965 + 0.3590084349i\)
\(L(1)\) \(\approx\) \(0.6603551965 + 0.3590084349i\)

Euler product

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

−34.61480111203549865752260874663, −33.58227919483154602615751645325, −31.877335689046355581458917017423, −30.17800701376258662949445326536, −29.59091876293576439950561775522, −28.96654594099520994699020470348, −27.25387571323338205977018841748, −25.96384633585835919981122582488, −25.257659007015509623813077316381, −23.92748036163812102851333689699, −22.238483152538161247394752145514, −20.89917085603887733904883024926, −19.45530800265856741268402216955, −18.80726133898896658320065346965, −17.30197422967764616939031579924, −16.775941041297409202257142239441, −14.24664543823765243587339508451, −13.130440657276665531200145919468, −11.725844076293417977040819977026, −10.26104378733710409258580834583, −8.97501963009031004521120379724, −7.31305247199816800727266678047, −6.3213604143417173938116238408, −3.107946651971263010221201775127, −1.44643056412635985582160308848, 2.444158638839448889293888696202, 5.10947782001000455003081293174, 6.3296563583367392518086166111, 8.51486375665073543892518496365, 9.59050271152605424222760583917, 10.33426627788428484192681114514, 12.286612794887876946930223536935, 14.49050971330321669148209288696, 15.37321869537416553516807337995, 16.77748760978067799356689358592, 17.51906626970687980730802815503, 19.15445271150432604234836017743, 20.43273101412197387879575396967, 21.586222299044942503364176391436, 22.87630140972969784904672893905, 24.98878818738886948443858611854, 25.375468235018114233102616286457, 26.62399560550103286622030143233, 27.998560976698507234100901033457, 28.454671427468243347198427784097, 29.88641126573817020740756206656, 32.086449736243884680616169741882, 32.62162335276769186292316343966, 33.84117810965259355426538692474, 34.70651914803993834814203265428

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