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Properties

Label 1-17e2-289.50-r0-0-0
Degree $1$
Conductor $289$
Sign $0.0597 + 0.998i$
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  + (−0.602 − 0.798i)2-s + (−0.739 + 0.673i)3-s + (−0.273 + 0.961i)4-s + (−0.932 − 0.361i)5-s + (0.982 + 0.183i)6-s + (−0.0922 + 0.995i)7-s + (0.932 − 0.361i)8-s + (0.0922 − 0.995i)9-s + (0.273 + 0.961i)10-s + (0.273 − 0.961i)11-s + (−0.445 − 0.895i)12-s + (0.932 − 0.361i)13-s + (0.850 − 0.526i)14-s + (0.932 − 0.361i)15-s + (−0.850 − 0.526i)16-s + ⋯
L(s)  = 1  + (−0.602 − 0.798i)2-s + (−0.739 + 0.673i)3-s + (−0.273 + 0.961i)4-s + (−0.932 − 0.361i)5-s + (0.982 + 0.183i)6-s + (−0.0922 + 0.995i)7-s + (0.932 − 0.361i)8-s + (0.0922 − 0.995i)9-s + (0.273 + 0.961i)10-s + (0.273 − 0.961i)11-s + (−0.445 − 0.895i)12-s + (0.932 − 0.361i)13-s + (0.850 − 0.526i)14-s + (0.932 − 0.361i)15-s + (−0.850 − 0.526i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0597 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0597 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.0597 + 0.998i$
Analytic conductor: \(1.34211\)
Root analytic conductor: \(1.34211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 289,\ (0:\ ),\ 0.0597 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2574550349 + 0.2425047592i\)
\(L(\frac12)\) \(\approx\) \(0.2574550349 + 0.2425047592i\)
\(L(1)\) \(\approx\) \(0.4813149889 + 0.01266897952i\)
\(L(1)\) \(\approx\) \(0.4813149889 + 0.01266897952i\)

Euler product

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

−25.38175830060689547302169177211, −24.14260272634916060845949843408, −23.52504715988229446269920863683, −23.06839961519895662994778105696, −22.20722358703545826873545462077, −20.27429488286970333277835933143, −19.647665958955957413082204301434, −18.65268661560000231391085198419, −18.005224367497341505859716516896, −16.98295239929493130510878448755, −16.35795712703342548799842523556, −15.36510699720081849164257556695, −14.3151362846885256475859472413, −13.31710813870924508684979764449, −12.164619215903682682046727186399, −10.88399388809692061307795694834, −10.5366348888072057315492335082, −8.909354213933609706468846104274, −7.81311711971681122298298541484, −6.93280945100959352772647154158, −6.57473848368102926567975845122, −4.9722251830709503800887079900, −4.002465032337985181063833825961, −1.78399362334393792982617867375, −0.36385084750190179424017018951, 1.26839893814552988330527426794, 3.22688873198469764625526829443, 3.8977511445901123207483150745, 5.214034150728978384067570965947, 6.37995867950692249161468963747, 8.10973089937319421836651780814, 8.72641085962632312608712318597, 9.74590931434059562569164746088, 10.93850919732411199024053859834, 11.56200576639365631108866762177, 12.232337992503280224945268910634, 13.270619493633625474479500368738, 14.99987202832140630060882004136, 15.97636302911017404192461020388, 16.4792917325950967269593541736, 17.5805347814739729347241303453, 18.60205595610409463860273300575, 19.23653379917376487037670829964, 20.392087569622772598503270866004, 21.21325040962800716773339298277, 21.90818513466325743504028591329, 22.82976984125369504592068568834, 23.663975220045239727254021933913, 24.967292323130877252254554286309, 25.982187588528094378972635004605

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