Properties

Label 1-297-297.47-r1-0-0
Degree $1$
Conductor $297$
Sign $-0.196 + 0.980i$
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  + (0.374 + 0.927i)2-s + (−0.719 + 0.694i)4-s + (0.615 + 0.788i)5-s + (0.438 − 0.898i)7-s + (−0.913 − 0.406i)8-s + (−0.5 + 0.866i)10-s + (0.848 − 0.529i)13-s + (0.997 + 0.0697i)14-s + (0.0348 − 0.999i)16-s + (0.978 + 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.990 − 0.139i)20-s + (−0.173 + 0.984i)23-s + (−0.241 + 0.970i)25-s + (0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (0.374 + 0.927i)2-s + (−0.719 + 0.694i)4-s + (0.615 + 0.788i)5-s + (0.438 − 0.898i)7-s + (−0.913 − 0.406i)8-s + (−0.5 + 0.866i)10-s + (0.848 − 0.529i)13-s + (0.997 + 0.0697i)14-s + (0.0348 − 0.999i)16-s + (0.978 + 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.990 − 0.139i)20-s + (−0.173 + 0.984i)23-s + (−0.241 + 0.970i)25-s + (0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ -0.196 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.758788002 + 2.145169854i\)
\(L(\frac12)\) \(\approx\) \(1.758788002 + 2.145169854i\)
\(L(1)\) \(\approx\) \(1.271696251 + 0.8520557056i\)
\(L(1)\) \(\approx\) \(1.271696251 + 0.8520557056i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.374 + 0.927i)T \)
5 \( 1 + (0.615 + 0.788i)T \)
7 \( 1 + (0.438 - 0.898i)T \)
13 \( 1 + (0.848 - 0.529i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.997 - 0.0697i)T \)
31 \( 1 + (-0.882 - 0.469i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (0.997 + 0.0697i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.719 + 0.694i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.961 - 0.275i)T \)
61 \( 1 + (-0.882 + 0.469i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (-0.374 - 0.927i)T \)
83 \( 1 + (-0.848 - 0.529i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.615 + 0.788i)T \)
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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

−24.76336172777267210716131748687, −23.980004333593016936909066978855, −23.05099211306247357630032255541, −21.87731649692674510592106944649, −21.32856360360316740618416474912, −20.60569303897923617591785038784, −19.74188402955590105874751799434, −18.37648940501499688528177670418, −18.169596893479238155100910763370, −16.74975697392351063499615274703, −15.71311735093532018187689433649, −14.429821147378494449144515328895, −13.78367918700388305213273024467, −12.680821764150838285854689107194, −12.017326768611002245450146076789, −11.069396000718527626429919914281, −9.84174564454641459034705627989, −9.05511189386349846611461112456, −8.24047633580620570308252363275, −6.24782969467556861391684980362, −5.3529365452086561159857415653, −4.54345608185865372687353784963, −3.09332906032642954880757462503, −1.91566072575146715154739590854, −0.92140446977299749376584399726, 1.16658944059835463415071256339, 3.08950519282966613663635441345, 3.979266670597759383491284698003, 5.42417048293278621802245767380, 6.142694092259132012995609258633, 7.37286364525998638780259644606, 7.92371036427491667071207106664, 9.39722313505654792532274478704, 10.346056713920396165648030321805, 11.440891047387162041977765885647, 12.81272233091207764517942179914, 13.8283699696625506941446245450, 14.22549284079140342668523055018, 15.262600085994294456815958030700, 16.28281847238809126103733528117, 17.23524103172733526538162569514, 17.96693428870474966646467780064, 18.69960693552614277266289690067, 20.213466508247288400076959865725, 21.20994195321338601443333244721, 21.94263073655427594402558686227, 23.138012055986063272713396163891, 23.34835541271330424395048937163, 24.64028957416093329987417297235, 25.47431050474459049353508159049

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