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Properties

Label 1-247-247.186-r1-0-0
Degree $1$
Conductor $247$
Sign $-0.107 - 0.994i$
Analytic cond. $26.5438$
Root an. cond. $26.5438$
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  + (0.173 − 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.173 + 0.984i)5-s + (−0.766 + 0.642i)6-s − 7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.939 + 0.342i)10-s − 11-s + (0.5 + 0.866i)12-s + (−0.173 + 0.984i)14-s + (0.766 − 0.642i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + 18-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.173 + 0.984i)5-s + (−0.766 + 0.642i)6-s − 7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.939 + 0.342i)10-s − 11-s + (0.5 + 0.866i)12-s + (−0.173 + 0.984i)14-s + (0.766 − 0.642i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $-0.107 - 0.994i$
Analytic conductor: \(26.5438\)
Root analytic conductor: \(26.5438\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{247} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (1:\ ),\ -0.107 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4209403082 - 0.4688498135i\)
\(L(\frac12)\) \(\approx\) \(0.4209403082 - 0.4688498135i\)
\(L(1)\) \(\approx\) \(0.5270765327 - 0.2951354593i\)
\(L(1)\) \(\approx\) \(0.5270765327 - 0.2951354593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−26.10590301830690337985302535500, −25.15814936262689615166140666016, −24.097460707832779278082160531350, −23.357774972169186818340175439286, −22.69498448534263698479681017309, −21.675630261431687504402602696, −20.87459756682685476500173504878, −19.59726082171432432877024492567, −18.2976032864743700097672810382, −17.346379024246953037178137525575, −16.58794545221188569149396255993, −15.6087121922401507415433128463, −15.574521453091761222292178422050, −13.71075316297557319851161297899, −12.86403021393636101063109468877, −12.093406671044190935053332210878, −10.56502524300463308381304931899, −9.45134674979790908822495859782, −8.7597451939213593446770682449, −7.34376267172862441079251504912, −6.243357014049738392569499505201, −5.25702127249582207286506858444, −4.53143016623258454269905045111, −3.303529050331071501921497431821, −0.56256438004885102510062677372, 0.4190185188517454079867712931, 2.2122814548644808460220685535, 3.071814398060560928039116645954, 4.52576081553439067718704784142, 5.86094581063335666815046632406, 6.74104250641186468798140558464, 7.99169714622353326755793570414, 9.52180943704905987950275078778, 10.72054974555077889286110824464, 10.98209887362701384525382225424, 12.35337110014176213069014571767, 12.99255616465413192710647675031, 13.85288834362569636647788342656, 15.16007325322742736688027470653, 16.303596515164562068360991668568, 17.695961294910961732363534954133, 18.29472032361272325580050108010, 19.160090267559854598606516829943, 19.71182714879785706701499380825, 21.18572886800386130039431738918, 22.12602347049890875957537974565, 22.79491437774078335794140846370, 23.32284036207697689426151040378, 24.38847024696484420156531643769, 25.87340972301505183999888427965

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