Properties

Label 1-43-43.39-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: $\chi_{43} (39, \cdot )$
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

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

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