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Properties

Label 1-17e2-289.84-r0-0-0
Degree $1$
Conductor $289$
Sign $-0.645 + 0.764i$
Analytic cond. $1.34211$
Root an. cond. $1.34211$
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  + (−0.850 + 0.526i)2-s + (0.982 − 0.183i)3-s + (0.445 − 0.895i)4-s + (−0.0922 + 0.995i)5-s + (−0.739 + 0.673i)6-s + (−0.932 + 0.361i)7-s + (0.0922 + 0.995i)8-s + (0.932 − 0.361i)9-s + (−0.445 − 0.895i)10-s + (−0.445 + 0.895i)11-s + (0.273 − 0.961i)12-s + (0.0922 + 0.995i)13-s + (0.602 − 0.798i)14-s + (0.0922 + 0.995i)15-s + (−0.602 − 0.798i)16-s + ⋯
L(s)  = 1  + (−0.850 + 0.526i)2-s + (0.982 − 0.183i)3-s + (0.445 − 0.895i)4-s + (−0.0922 + 0.995i)5-s + (−0.739 + 0.673i)6-s + (−0.932 + 0.361i)7-s + (0.0922 + 0.995i)8-s + (0.932 − 0.361i)9-s + (−0.445 − 0.895i)10-s + (−0.445 + 0.895i)11-s + (0.273 − 0.961i)12-s + (0.0922 + 0.995i)13-s + (0.602 − 0.798i)14-s + (0.0922 + 0.995i)15-s + (−0.602 − 0.798i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.645 + 0.764i$
Analytic conductor: \(1.34211\)
Root analytic conductor: \(1.34211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 289,\ (0:\ ),\ -0.645 + 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3472145445 + 0.7475486166i\)
\(L(\frac12)\) \(\approx\) \(0.3472145445 + 0.7475486166i\)
\(L(1)\) \(\approx\) \(0.7053251066 + 0.4106812875i\)
\(L(1)\) \(\approx\) \(0.7053251066 + 0.4106812875i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + (-0.850 + 0.526i)T \)
3 \( 1 + (0.982 - 0.183i)T \)
5 \( 1 + (-0.0922 + 0.995i)T \)
7 \( 1 + (-0.932 + 0.361i)T \)
11 \( 1 + (-0.445 + 0.895i)T \)
13 \( 1 + (0.0922 + 0.995i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (-0.932 + 0.361i)T \)
29 \( 1 + (-0.445 + 0.895i)T \)
31 \( 1 + (-0.0922 - 0.995i)T \)
37 \( 1 + (0.273 + 0.961i)T \)
41 \( 1 + (0.982 - 0.183i)T \)
43 \( 1 + (-0.602 + 0.798i)T \)
47 \( 1 + (0.932 + 0.361i)T \)
53 \( 1 + (0.932 - 0.361i)T \)
59 \( 1 + (0.739 + 0.673i)T \)
61 \( 1 + (-0.739 + 0.673i)T \)
67 \( 1 + (-0.850 - 0.526i)T \)
71 \( 1 + (-0.932 + 0.361i)T \)
73 \( 1 + (0.602 + 0.798i)T \)
79 \( 1 + (0.850 + 0.526i)T \)
83 \( 1 + (-0.982 - 0.183i)T \)
89 \( 1 + (0.0922 - 0.995i)T \)
97 \( 1 + (-0.932 + 0.361i)T \)
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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

−25.24690951123383507495726990924, −24.76957960280359456076925472701, −23.51780850234988306490177856604, −22.10775743193894275123288409424, −21.15419028139021196317190550138, −20.48819347481812852670805036942, −19.70538095774527043168969495727, −19.15783740740587125036519848507, −18.08066444519827942967794378826, −16.78160832581683782804515288231, −16.1578108029989032652327007564, −15.4066317473591733851417406730, −13.7420303508843021385216210233, −12.967219658538449271136256164, −12.3192222547379052011685984635, −10.68328605859697339781944551333, −10.00597419286178584510457156259, −8.98533000295126557207819688618, −8.295034973739954373372742966186, −7.49841711879958295274794828046, −5.941022755826058500901652116220, −4.1408595945356062918172999517, −3.33806755546060206602906680441, −2.15758877483145508272599010999, −0.610507063883627508990705490296, 1.96267614277946439464226285699, 2.72988039784193597872531608046, 4.19726615119298849306011669950, 6.08132138811072864658527822185, 6.92733309615603753596855220805, 7.60000903270683257255297927421, 8.82779302940035227543875580268, 9.65774918735420880598693101554, 10.35610775889896485767844438705, 11.68401113628719746689057812889, 13.03756116583234944413386917522, 14.10728375759175866426382075184, 15.023050051033845560487092734305, 15.52447443090082170497103265064, 16.575005714279506958663569093057, 17.98892454254179880490230617614, 18.56297307843497328884705860361, 19.3481307478932718596095632415, 19.94719021124574586798506695387, 21.15832799779020791360219805476, 22.28383936684022273719066939413, 23.42715528050030001496367220294, 24.166670166614143486962095454061, 25.47177453148400507884681706043, 25.92133583576762393099648038513

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