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Properties

Label 1-17e2-289.54-r1-0-0
Degree $1$
Conductor $289$
Sign $0.677 - 0.735i$
Analytic cond. $31.0573$
Root an. cond. $31.0573$
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  + (0.403 − 0.914i)2-s + (−0.251 − 0.967i)3-s + (−0.673 − 0.739i)4-s + (−0.997 + 0.0692i)5-s + (−0.986 − 0.160i)6-s + (0.993 − 0.115i)7-s + (−0.948 + 0.317i)8-s + (−0.873 + 0.486i)9-s + (−0.339 + 0.940i)10-s + (−0.424 + 0.905i)11-s + (−0.545 + 0.837i)12-s + (0.895 + 0.445i)13-s + (0.295 − 0.955i)14-s + (0.317 + 0.948i)15-s + (−0.0922 + 0.995i)16-s + ⋯
L(s)  = 1  + (0.403 − 0.914i)2-s + (−0.251 − 0.967i)3-s + (−0.673 − 0.739i)4-s + (−0.997 + 0.0692i)5-s + (−0.986 − 0.160i)6-s + (0.993 − 0.115i)7-s + (−0.948 + 0.317i)8-s + (−0.873 + 0.486i)9-s + (−0.339 + 0.940i)10-s + (−0.424 + 0.905i)11-s + (−0.545 + 0.837i)12-s + (0.895 + 0.445i)13-s + (0.295 − 0.955i)14-s + (0.317 + 0.948i)15-s + (−0.0922 + 0.995i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.677 - 0.735i$
Analytic conductor: \(31.0573\)
Root analytic conductor: \(31.0573\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 289,\ (1:\ ),\ 0.677 - 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267521305 - 0.5555540854i\)
\(L(\frac12)\) \(\approx\) \(1.267521305 - 0.5555540854i\)
\(L(1)\) \(\approx\) \(0.8096988642 - 0.5751767938i\)
\(L(1)\) \(\approx\) \(0.8096988642 - 0.5751767938i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + (0.403 - 0.914i)T \)
3 \( 1 + (-0.251 - 0.967i)T \)
5 \( 1 + (-0.997 + 0.0692i)T \)
7 \( 1 + (0.993 - 0.115i)T \)
11 \( 1 + (-0.424 + 0.905i)T \)
13 \( 1 + (0.895 + 0.445i)T \)
19 \( 1 + (0.914 - 0.403i)T \)
23 \( 1 + (0.115 + 0.993i)T \)
29 \( 1 + (-0.339 - 0.940i)T \)
31 \( 1 + (-0.754 + 0.656i)T \)
37 \( 1 + (-0.978 - 0.206i)T \)
41 \( 1 + (-0.506 + 0.862i)T \)
43 \( 1 + (-0.638 + 0.769i)T \)
47 \( 1 + (0.961 - 0.273i)T \)
53 \( 1 + (0.486 + 0.873i)T \)
59 \( 1 + (0.973 + 0.228i)T \)
61 \( 1 + (0.160 - 0.986i)T \)
67 \( 1 + (-0.932 - 0.361i)T \)
71 \( 1 + (-0.620 + 0.783i)T \)
73 \( 1 + (0.884 + 0.466i)T \)
79 \( 1 + (-0.999 + 0.0230i)T \)
83 \( 1 + (-0.138 - 0.990i)T \)
89 \( 1 + (0.895 - 0.445i)T \)
97 \( 1 + (0.783 + 0.620i)T \)
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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

−25.430707905560842033752433222326, −24.14129406166547972734487967569, −23.78055880500058237372284096463, −22.67830312193225484379711390917, −22.09163123644530747598911656036, −20.85870712345352739469858029156, −20.48987469323287219050361344710, −18.71896162917913231377964743518, −17.96417251330474268547061812159, −16.71380455933749698731792275003, −16.142180501681479518466224720536, −15.35032262890406116710206340667, −14.62746062426814077549812213265, −13.6649186566916708970751497714, −12.26126765156680345950686722933, −11.376873281655455876628696338587, −10.51839446996278572018949780673, −8.74709754705537652382690322476, −8.416316168127227575896667479507, −7.25316412826737528002802649240, −5.726672882444491588967798548823, −5.09571069854931870374015448120, −3.9621834158565547108918645623, −3.23154254777081368643803036697, −0.479900223934039398901662317194, 1.0231141573239102678653658088, 1.98121202711076772804801646601, 3.37155717201125943796157540722, 4.59188145550783466957807594707, 5.51982309303929715004674151517, 7.03915932823423666769183882832, 7.92657131648387660709156469259, 8.97283202835782761695476541869, 10.54424894616380416176988939491, 11.528952476815616454277889028513, 11.79790071863467512769906808715, 12.96404476456032337019556296860, 13.79233093449847699680296226683, 14.76081813958668635754168049306, 15.75822219204385875550975174768, 17.3471598974027385943182194847, 18.19713903356804253662274280940, 18.768675957233836630328312275977, 19.881695338502891736009922485500, 20.39343900563222158224196798384, 21.48137407972675128531177732185, 22.72725660814196315825671472561, 23.386071179222680296069781604578, 23.85061245565887106173990871231, 24.79506451931809939260239527490

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