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Properties

Label 1-297-297.5-r1-0-0
Degree $1$
Conductor $297$
Sign $0.773 - 0.633i$
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  + (−0.438 − 0.898i)2-s + (−0.615 + 0.788i)4-s + (0.997 − 0.0697i)5-s + (0.0348 + 0.999i)7-s + (0.978 + 0.207i)8-s + (−0.5 − 0.866i)10-s + (−0.719 − 0.694i)13-s + (0.882 − 0.469i)14-s + (−0.241 − 0.970i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.559 + 0.829i)20-s + (0.939 + 0.342i)23-s + (0.990 − 0.139i)25-s + (−0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (−0.438 − 0.898i)2-s + (−0.615 + 0.788i)4-s + (0.997 − 0.0697i)5-s + (0.0348 + 0.999i)7-s + (0.978 + 0.207i)8-s + (−0.5 − 0.866i)10-s + (−0.719 − 0.694i)13-s + (0.882 − 0.469i)14-s + (−0.241 − 0.970i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.559 + 0.829i)20-s + (0.939 + 0.342i)23-s + (0.990 − 0.139i)25-s + (−0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.773 - 0.633i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ 0.773 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.619575391 - 0.5781146503i\)
\(L(\frac12)\) \(\approx\) \(1.619575391 - 0.5781146503i\)
\(L(1)\) \(\approx\) \(0.9802024320 - 0.3021972866i\)
\(L(1)\) \(\approx\) \(0.9802024320 - 0.3021972866i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.438 - 0.898i)T \)
5 \( 1 + (0.997 - 0.0697i)T \)
7 \( 1 + (0.0348 + 0.999i)T \)
13 \( 1 + (-0.719 - 0.694i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.882 + 0.469i)T \)
31 \( 1 + (0.961 - 0.275i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (0.882 - 0.469i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.615 + 0.788i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.374 + 0.927i)T \)
61 \( 1 + (0.961 + 0.275i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (0.719 - 0.694i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.997 - 0.0697i)T \)
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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

−25.28562987008804898055895921888, −24.53482392438859082527008955700, −23.61299074850253192800648019806, −22.862881796420498238154785187919, −21.72500243251106725943626722841, −20.86923041484246967665450507441, −19.51886398468249645596147641458, −18.92449110783468040737774867046, −17.52107691749399998067072371249, −17.218901998488284612684563178954, −16.44593549770437243822968408370, −15.10791856586595834171727730881, −14.27298153380247473370904326362, −13.61414450358554310291135034329, −12.57314164142321322557058760722, −10.73042783801917413808451880993, −10.203199535562699991780681406906, −9.18871006498229346205850268366, −8.179446106158958342024670346393, −6.93607951915797617161158420743, −6.360378055680905010025924929934, −5.09333519573588478072647216177, −4.10175188798690577825014662514, −2.130265057820473529964857233551, −0.85122573375386235828033733207, 0.88364326955747619488604972637, 2.33607164219935518968702737522, 2.8766670969036918345703440367, 4.6933902393875354964888696962, 5.562239804774911918449382449117, 7.00968651016960880866677641621, 8.39952665843829454351563732605, 9.19924029383661627343463713000, 9.99035689819610349932980678829, 10.95742759878465850860825751741, 12.12437120548638445470995356526, 12.80169493045936507679337782456, 13.773230583137140908104300230499, 14.8463450478548662056330519966, 16.11421899532470265171612181827, 17.37971555681948551606686765497, 17.75257370081522503016228451906, 18.816309280558581028889200314012, 19.541854868485306035034832769314, 20.83159996693350276029750076580, 21.25832839929011070912189686457, 22.20132835204506802076635613122, 22.84209759545762747928414265607, 24.48782343345041675076634723914, 25.27260757865556452789424500841

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