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Properties

Label 1-3920-3920.69-r1-0-0
Degree $1$
Conductor $3920$
Sign $0.337 - 0.941i$
Analytic cond. $421.262$
Root an. cond. $421.262$
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  + (0.433 + 0.900i)3-s + (−0.623 + 0.781i)9-s + (−0.781 + 0.623i)11-s + (0.781 − 0.623i)13-s + (−0.222 − 0.974i)17-s − i·19-s + (−0.222 + 0.974i)23-s + (−0.974 − 0.222i)27-s + (−0.974 + 0.222i)29-s − 31-s + (−0.900 − 0.433i)33-s + (−0.974 + 0.222i)37-s + (0.900 + 0.433i)39-s + (−0.900 + 0.433i)41-s + (−0.433 + 0.900i)43-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)3-s + (−0.623 + 0.781i)9-s + (−0.781 + 0.623i)11-s + (0.781 − 0.623i)13-s + (−0.222 − 0.974i)17-s − i·19-s + (−0.222 + 0.974i)23-s + (−0.974 − 0.222i)27-s + (−0.974 + 0.222i)29-s − 31-s + (−0.900 − 0.433i)33-s + (−0.974 + 0.222i)37-s + (0.900 + 0.433i)39-s + (−0.900 + 0.433i)41-s + (−0.433 + 0.900i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.337 - 0.941i$
Analytic conductor: \(421.262\)
Root analytic conductor: \(421.262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3920,\ (1:\ ),\ 0.337 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09581349562 - 0.06740770559i\)
\(L(\frac12)\) \(\approx\) \(0.09581349562 - 0.06740770559i\)
\(L(1)\) \(\approx\) \(0.8861875677 + 0.4023460168i\)
\(L(1)\) \(\approx\) \(0.8861875677 + 0.4023460168i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.433 + 0.900i)T \)
11 \( 1 + (-0.781 + 0.623i)T \)
13 \( 1 + (0.781 - 0.623i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 - iT \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (-0.974 + 0.222i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.974 + 0.222i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.974 + 0.222i)T \)
59 \( 1 + (-0.433 + 0.900i)T \)
61 \( 1 + (0.974 - 0.222i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.781 - 0.623i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + T \)
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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

−18.63874372155173325098367018333, −17.99894959796931286411948514625, −17.158990787132017255822243441080, −16.56566231396445317210506449059, −15.63405170261390345459068966260, −15.07415993631154151391907617116, −14.24341620859294275557043985614, −13.55915306053111331693836641848, −13.15196678383936603572644585909, −12.45311282955087331300835962291, −11.634434318148562730108905047798, −10.945999171949204380126004000086, −10.32208580736490824525084484792, −9.091002877822925673889061449074, −8.68796167494753667477765458385, −8.101625230757387605009031530002, −7.17130390324206313415095631407, −6.63764056101157381536548657974, −5.84945465400686348123422118406, −5.14423280477960111276756581754, −3.87852967774239494223867100528, −3.4190893382336314136004352056, −2.263610323346148579454254369874, −1.87469268210388825159245613229, −0.69307740073022192313199728970, 0.019800541504800209845923970064, 1.444969375669908765907207756967, 2.30346951710677532127526144107, 3.23747498210416979724126300741, 3.72057204533359170237068024448, 4.68120982918320121156636777107, 5.38021269027046021223355017771, 5.905557983853232769410442565231, 7.22233724679228506150473842611, 7.739185898362852038285184367393, 8.52768903054700653195105088785, 9.22600871771740650163083878616, 9.95902417818738473963521009770, 10.45538329708197837028284090976, 11.20973133629992171941473569394, 11.89552226516425516186682835434, 12.95448192847172213366331062377, 13.42770260408799100883934394427, 14.21779543673172139640362545655, 14.928359476591454851503286805933, 15.53204418933743480088879539375, 16.052718663549745285635853418738, 16.67145264020712037882370376824, 17.548922054035818736085310867268, 18.2782910373057167242038929119

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