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Properties

Label 1-43-43.32-r1-0-0
Degree $1$
Conductor $43$
Sign $-0.335 + 0.941i$
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 + (0.222 + 0.974i)3-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)5-s + 6-s − 7-s + (−0.623 + 0.781i)8-s + (−0.900 + 0.433i)9-s + (0.623 + 0.781i)10-s + (−0.900 + 0.433i)11-s + (0.222 − 0.974i)12-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)14-s + (−0.900 − 0.433i)15-s + (0.623 + 0.781i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (0.222 + 0.974i)3-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)5-s + 6-s − 7-s + (−0.623 + 0.781i)8-s + (−0.900 + 0.433i)9-s + (0.623 + 0.781i)10-s + (−0.900 + 0.433i)11-s + (0.222 − 0.974i)12-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)14-s + (−0.900 − 0.433i)15-s + (0.623 + 0.781i)16-s + (0.623 + 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.335 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.335 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $-0.335 + 0.941i$
Analytic conductor: \(4.62099\)
Root analytic conductor: \(4.62099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (1:\ ),\ -0.335 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3842833496 + 0.5450649864i\)
\(L(\frac12)\) \(\approx\) \(0.3842833496 + 0.5450649864i\)
\(L(1)\) \(\approx\) \(0.7630349434 + 0.09062746632i\)
\(L(1)\) \(\approx\) \(0.7630349434 + 0.09062746632i\)

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 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 + (0.900 + 0.433i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (-0.222 + 0.974i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
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

−34.26648299667169460655313228999, −32.636932188738752859930886134737, −31.64066869157166862385094253209, −30.998673193285602234086024572039, −29.31615663215107160135232788562, −28.14260163195966070779397015796, −26.43647841943809506648207278682, −25.59496430146471831253059281115, −24.288788723494376448693239164117, −23.65265845956530313863665509496, −22.55668020881458243365400494006, −20.65728527731626550799643171632, −19.15467718876295440785971647151, −18.21914738578556653382655413211, −16.522880631823384924308301231387, −15.82620570070164009812382255290, −13.96124917575843859610261189607, −13.0570625715652595594149608658, −11.96736715672083879842066659036, −9.24272644931622167890735824299, −8.08338727441646031102943420186, −6.89901878442572306815374931099, −5.442679668166386640173562115621, −3.4280900801474490839843263913, −0.36865450530671526548717449797, 2.94633748423360130832718194226, 3.80846726493203477542885346333, 5.65888408957079633347008940541, 8.14021100367908462797598963084, 9.94540547624920890662014567292, 10.52519028729196426382241541134, 12.038302161695688326284492153048, 13.57670481028663867829265970892, 15.000983617005284137253892497253, 15.99312537145422952591305342657, 18.04178169351752176566286323851, 19.33410133817386393488949423367, 20.26525483832826563197244160294, 21.50443542023793556983660394174, 22.67395042522579101414970137881, 23.23072785352214772504646066148, 25.788644540927348465021036145333, 26.59437048576259269321122334023, 27.789654572067394396368458058548, 28.71077310910803284978835904572, 30.164164093105894330608835814476, 31.2286407304953377495842283090, 32.14600967723580680619090074279, 33.19387908673110704517448604082, 34.686226325180426754804052224976

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