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Properties

Label 1-297-297.104-r1-0-0
Degree $1$
Conductor $297$
Sign $-0.935 - 0.353i$
Analytic cond. $31.9170$
Root an. cond. $31.9170$
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  + (0.997 + 0.0697i)2-s + (0.990 + 0.139i)4-s + (−0.559 − 0.829i)5-s + (−0.882 − 0.469i)7-s + (0.978 + 0.207i)8-s + (−0.5 − 0.866i)10-s + (−0.241 + 0.970i)13-s + (−0.848 − 0.529i)14-s + (0.961 + 0.275i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.438 − 0.898i)20-s + (−0.173 − 0.984i)23-s + (−0.374 + 0.927i)25-s + (−0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (0.997 + 0.0697i)2-s + (0.990 + 0.139i)4-s + (−0.559 − 0.829i)5-s + (−0.882 − 0.469i)7-s + (0.978 + 0.207i)8-s + (−0.5 − 0.866i)10-s + (−0.241 + 0.970i)13-s + (−0.848 − 0.529i)14-s + (0.961 + 0.275i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.438 − 0.898i)20-s + (−0.173 − 0.984i)23-s + (−0.374 + 0.927i)25-s + (−0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.935 - 0.353i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ -0.935 - 0.353i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2001338365 - 1.095206679i\)
\(L(\frac12)\) \(\approx\) \(0.2001338365 - 1.095206679i\)
\(L(1)\) \(\approx\) \(1.273932653 - 0.3098410539i\)
\(L(1)\) \(\approx\) \(1.273932653 - 0.3098410539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.997 + 0.0697i)T \)
5 \( 1 + (-0.559 - 0.829i)T \)
7 \( 1 + (-0.882 - 0.469i)T \)
13 \( 1 + (-0.241 + 0.970i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.848 + 0.529i)T \)
31 \( 1 + (-0.719 - 0.694i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (-0.848 - 0.529i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.990 + 0.139i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.615 - 0.788i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (-0.997 - 0.0697i)T \)
83 \( 1 + (0.241 + 0.970i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.559 - 0.829i)T \)
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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

−25.58834405720591688116027602598, −24.59317673663992480447627829620, −23.42152534413102738744701082291, −22.95582243292655064082891019513, −22.01985919575094803536219773996, −21.48280262057950789289489496799, −20.05731987421945032435079552653, −19.45170857976806010795849553338, −18.63466002311540059144544379995, −17.230433042881265832654673664750, −16.08643919047552279156955324972, −15.17840426924513433581128848246, −14.82882135670111238802281323903, −13.45020457901365820842713615796, −12.65489869063939714652498355083, −11.82098289817484841809020672292, −10.72062411896480121802633386417, −10.00696063048621117533326697985, −8.296778214300253155340542201614, −7.19362377991872699272599544822, −6.282508130349410503294633387000, −5.39476374007650489013282982782, −3.81720148035487870555388799896, −3.2300590342978274870764646044, −2.01658560024298768453738074041, 0.211221145516107481712770184442, 1.93661973486445211099030682363, 3.398824808941362984134530012795, 4.28721270379816592983872050371, 5.18214730884622470204227167375, 6.55539901732944886309109534395, 7.26718950207952818501140729629, 8.58543732161475938216569496692, 9.76619627091428163580430600275, 11.04726958631728517076141574217, 11.98465347463672203598388359409, 12.79495092158720180919082738647, 13.53290253528884057919233799184, 14.595324661610581744545510957621, 15.631618934721510484887430957958, 16.56974870917236234535136714565, 16.82189102303543408907733949601, 18.74196223174215317939172595094, 19.65688285167306006275259328131, 20.36663734727636453595412420325, 21.16798857077605590521313692144, 22.2892010806613399798691273758, 23.00853956671775414056833012151, 23.87963330814213282090665713582, 24.44806397061626466077515334301

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