Properties

Label 1-43-43.30-r1-0-0
Degree $1$
Conductor $43$
Sign $-0.503 + 0.864i$
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  + (0.900 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $-0.503 + 0.864i$
Analytic conductor: \(4.62099\)
Root analytic conductor: \(4.62099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (1:\ ),\ -0.503 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.055024622 + 1.835593959i\)
\(L(\frac12)\) \(\approx\) \(1.055024622 + 1.835593959i\)
\(L(1)\) \(\approx\) \(1.229810082 + 0.9898779194i\)
\(L(1)\) \(\approx\) \(1.229810082 + 0.9898779194i\)

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

−33.97550135264056635260129101293, −32.60451954721551336237255209603, −31.24027129156225389800731709504, −30.40825038527799884094897781277, −29.81676953630929058345173397766, −28.36266031607359635593043741565, −27.08788275201073927866129340006, −25.20460784476470660795601362024, −24.09683106736943140864086796003, −23.19826051043562525974551567885, −22.42343011226191406651909200181, −20.34368714299720118188396206952, −19.77970036300056360309336359285, −18.463615887409058326352229699936, −16.83776710239368500031202517026, −14.94478900325879985354419080322, −14.097424537131946220508542578858, −12.53851355429131882417369482467, −11.76431731346451045224684145901, −10.404310862759040133167272216772, −7.7520808488237970527744751238, −6.81438443460058307245429856469, −4.8533363327096132208105592679, −3.17326300015674865775396292630, −1.13876102394738274083525472572, 3.18257907924013307436748018584, 4.52985905401191320022209703182, 5.69035964075830544218634997858, 7.76067688800129770689720361133, 9.14308230386593759811049475898, 11.489502316175733790026973955713, 11.91847849506780059768347309850, 14.08959363763419881733290868210, 15.13210195157242102212897073672, 16.06307316678443873114637211200, 17.08245772728451635125580454426, 19.31752266567367975739819342977, 20.74913312661126870608916776904, 21.693754131246465465193212890446, 22.68472416017094808108271005853, 23.98356825726259237943844789240, 25.00626481729132838543650005089, 26.59702503731744895018753914068, 27.47015190747235562510343845671, 28.81753345635915390353954695474, 30.51278555348308758786832462141, 31.77024386347701548440781716266, 32.01769481757096341534737153695, 33.580995004895736549319876157663, 34.43905359629817189293051961685

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