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Properties

Label 1-417-417.101-r0-0-0
Degree $1$
Conductor $417$
Sign $0.841 + 0.539i$
Analytic cond. $1.93653$
Root an. cond. $1.93653$
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  + (−0.949 + 0.313i)2-s + (0.803 − 0.595i)4-s + (0.648 + 0.761i)5-s + (−0.974 + 0.225i)7-s + (−0.576 + 0.816i)8-s + (−0.854 − 0.519i)10-s + (−0.613 − 0.789i)11-s + (0.0227 − 0.999i)13-s + (0.854 − 0.519i)14-s + (0.291 − 0.956i)16-s + (0.898 + 0.439i)17-s + (0.829 − 0.557i)19-s + (0.974 + 0.225i)20-s + (0.829 + 0.557i)22-s + (0.682 + 0.730i)23-s + ⋯
L(s)  = 1  + (−0.949 + 0.313i)2-s + (0.803 − 0.595i)4-s + (0.648 + 0.761i)5-s + (−0.974 + 0.225i)7-s + (−0.576 + 0.816i)8-s + (−0.854 − 0.519i)10-s + (−0.613 − 0.789i)11-s + (0.0227 − 0.999i)13-s + (0.854 − 0.519i)14-s + (0.291 − 0.956i)16-s + (0.898 + 0.439i)17-s + (0.829 − 0.557i)19-s + (0.974 + 0.225i)20-s + (0.829 + 0.557i)22-s + (0.682 + 0.730i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(417\)    =    \(3 \cdot 139\)
Sign: $0.841 + 0.539i$
Analytic conductor: \(1.93653\)
Root analytic conductor: \(1.93653\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{417} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 417,\ (0:\ ),\ 0.841 + 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8269438217 + 0.2422013633i\)
\(L(\frac12)\) \(\approx\) \(0.8269438217 + 0.2422013633i\)
\(L(1)\) \(\approx\) \(0.7395852525 + 0.1460968448i\)
\(L(1)\) \(\approx\) \(0.7395852525 + 0.1460968448i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.949 + 0.313i)T \)
5 \( 1 + (0.648 + 0.761i)T \)
7 \( 1 + (-0.974 + 0.225i)T \)
11 \( 1 + (-0.613 - 0.789i)T \)
13 \( 1 + (0.0227 - 0.999i)T \)
17 \( 1 + (0.898 + 0.439i)T \)
19 \( 1 + (0.829 - 0.557i)T \)
23 \( 1 + (0.682 + 0.730i)T \)
29 \( 1 + (-0.113 - 0.993i)T \)
31 \( 1 + (0.538 + 0.842i)T \)
37 \( 1 + (0.934 - 0.356i)T \)
41 \( 1 + (0.715 - 0.699i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.983 - 0.181i)T \)
53 \( 1 + (0.746 + 0.665i)T \)
59 \( 1 + (-0.917 + 0.398i)T \)
61 \( 1 + (0.715 + 0.699i)T \)
67 \( 1 + (0.377 + 0.926i)T \)
71 \( 1 + (-0.983 + 0.181i)T \)
73 \( 1 + (-0.995 - 0.0909i)T \)
79 \( 1 + (0.962 - 0.269i)T \)
83 \( 1 + (-0.377 + 0.926i)T \)
89 \( 1 + (0.158 + 0.987i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−24.44892405914081958882450364606, −23.34924525676034242949767376392, −22.298438672795276282431953562870, −21.21665536003162150202568098191, −20.6188538444919454501489953051, −19.9126049818785222876512448955, −18.816341861445657883064980076842, −18.23769067747880323536806317182, −17.08972377421470234458509684760, −16.470149732403146407681834251358, −15.91189564045510795573107744305, −14.49423054086105027247735544525, −13.23819202115457381793670313331, −12.56873172496882734397904008200, −11.72450532582127110211384173310, −10.37026207507506503818707726722, −9.675898470916414895618447845079, −9.15307144405526174368645780871, −7.91476320654547901721491826913, −6.96921695157665343680892975777, −5.97539058601298913958106198293, −4.64570433932466459260823075027, −3.2384254840430700115900838078, −2.135111929788994305215928302749, −0.92924474428803654107076987834, 0.97231466352633272873071480350, 2.67948407974229418940277987254, 3.17247862269230231963924378572, 5.571512102597029177875675986309, 5.93023522025610722281156139765, 7.1014360260757153039190506360, 7.92902716575570705892494032634, 9.13688055875226500024261458238, 9.94331565619069680734464839997, 10.58296105902785393276901839415, 11.54644759728827227295743981262, 12.90312712986627298821240320888, 13.79898771763297651979255850232, 14.93238912509998404821685743231, 15.68398126355879477960601549720, 16.48414446759058170138556029769, 17.53914697713299066957233314477, 18.1727817280743997911669105448, 19.09070618296489814951692710093, 19.56948372757820781584913564401, 20.87468828540253351742912872898, 21.616110322062705668378769420975, 22.72360792530665457351138329373, 23.44540860608442281263028933320, 24.76932058432734690530910014487

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