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Properties

Label 1-43-43.28-r1-0-0
Degree $1$
Conductor $43$
Sign $-0.528 + 0.849i$
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 + (0.733 + 0.680i)3-s + (−0.900 + 0.433i)4-s + (0.988 − 0.149i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s + (−0.623 − 0.781i)8-s + (0.0747 + 0.997i)9-s + (0.365 + 0.930i)10-s + (−0.900 − 0.433i)11-s + (−0.955 − 0.294i)12-s + (0.365 − 0.930i)13-s + (−0.733 + 0.680i)14-s + (0.826 + 0.563i)15-s + (0.623 − 0.781i)16-s + (−0.988 − 0.149i)17-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (0.733 + 0.680i)3-s + (−0.900 + 0.433i)4-s + (0.988 − 0.149i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s + (−0.623 − 0.781i)8-s + (0.0747 + 0.997i)9-s + (0.365 + 0.930i)10-s + (−0.900 − 0.433i)11-s + (−0.955 − 0.294i)12-s + (0.365 − 0.930i)13-s + (−0.733 + 0.680i)14-s + (0.826 + 0.563i)15-s + (0.623 − 0.781i)16-s + (−0.988 − 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $-0.528 + 0.849i$
Analytic conductor: \(4.62099\)
Root analytic conductor: \(4.62099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (1:\ ),\ -0.528 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.039000357 + 1.870314916i\)
\(L(\frac12)\) \(\approx\) \(1.039000357 + 1.870314916i\)
\(L(1)\) \(\approx\) \(1.138479997 + 1.068170652i\)
\(L(1)\) \(\approx\) \(1.138479997 + 1.068170652i\)

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 \( 1 + (0.733 + 0.680i)T \)
5 \( 1 + (0.988 - 0.149i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (-0.0747 + 0.997i)T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (0.733 - 0.680i)T \)
31 \( 1 + (0.955 + 0.294i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.365 + 0.930i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.955 + 0.294i)T \)
67 \( 1 + (0.0747 - 0.997i)T \)
71 \( 1 + (-0.826 - 0.563i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.733 - 0.680i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
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

−33.61447811782380796745168485581, −32.61542206679882215395502405909, −31.165751498402830373388937589304, −30.48807795601728321947799848292, −29.38704283600134676455948790657, −28.600429136199235081493703702947, −26.706622569040551280864756312048, −25.9060507496510848987346962684, −24.21434015021515889300057344785, −23.28771340618910574677542109459, −21.52208059668733857762425899887, −20.71067070381247797907616111986, −19.62582107927243046975251218989, −18.26090104368266206304974213313, −17.48824073694164205884394351781, −14.86423767678106270974487087465, −13.58314225939262311921936490889, −13.1955964841092784332895825010, −11.30769771766143456561244948815, −9.93185970487058287975675053032, −8.62931786979939522787639222163, −6.764899792798023816793709926143, −4.656481643519108471259534192695, −2.714593417798033872804888090223, −1.416464919459978516800292884982, 2.74756339275356225150125047438, 4.82631451205167455499516683737, 5.90550448655017176931107125992, 8.10293386957978787890148036719, 8.9666514293856065685379072983, 10.40935927418326508627767316185, 12.90565525544803281800616964577, 13.95143228088231139383543869488, 15.12360677369120540141585998672, 16.06111537538561166329495992323, 17.57038407906443311909501143980, 18.67203036419852389758702777551, 20.82578347356915059629250621464, 21.51150602432912735261096522992, 22.71284215138197861975770138612, 24.63742713509472512675689430778, 25.109434718705805660012714010283, 26.27888910008318763151808799078, 27.31319510397932501527000750125, 28.64138998128097421620489833058, 30.57133963963832061191638793338, 31.58691907828081542894374028074, 32.49327346632725495993134870051, 33.48713035137787452249750033594, 34.35658180309107827850350226181

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