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Properties

Label 1-3920-3920.93-r1-0-0
Degree $1$
Conductor $3920$
Sign $-0.881 - 0.471i$
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  + (−0.365 − 0.930i)3-s + (−0.733 + 0.680i)9-s + (−0.680 + 0.733i)11-s + (0.222 − 0.974i)13-s + (−0.997 − 0.0747i)17-s + (−0.866 + 0.5i)19-s + (0.997 − 0.0747i)23-s + (0.900 + 0.433i)27-s + (0.433 + 0.900i)29-s + (−0.5 + 0.866i)31-s + (0.930 + 0.365i)33-s + (−0.826 − 0.563i)37-s + (−0.988 + 0.149i)39-s + (−0.623 + 0.781i)41-s + (0.623 + 0.781i)43-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)3-s + (−0.733 + 0.680i)9-s + (−0.680 + 0.733i)11-s + (0.222 − 0.974i)13-s + (−0.997 − 0.0747i)17-s + (−0.866 + 0.5i)19-s + (0.997 − 0.0747i)23-s + (0.900 + 0.433i)27-s + (0.433 + 0.900i)29-s + (−0.5 + 0.866i)31-s + (0.930 + 0.365i)33-s + (−0.826 − 0.563i)37-s + (−0.988 + 0.149i)39-s + (−0.623 + 0.781i)41-s + (0.623 + 0.781i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.881 - 0.471i$
Analytic conductor: \(421.262\)
Root analytic conductor: \(421.262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3920,\ (1:\ ),\ -0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09607419915 - 0.3834863152i\)
\(L(\frac12)\) \(\approx\) \(0.09607419915 - 0.3834863152i\)
\(L(1)\) \(\approx\) \(0.7395720796 - 0.1544639718i\)
\(L(1)\) \(\approx\) \(0.7395720796 - 0.1544639718i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.365 - 0.930i)T \)
11 \( 1 + (-0.680 + 0.733i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 + (-0.997 - 0.0747i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.997 - 0.0747i)T \)
29 \( 1 + (0.433 + 0.900i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (-0.294 - 0.955i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (-0.149 - 0.988i)T \)
61 \( 1 + (-0.563 + 0.826i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (0.294 - 0.955i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 - iT \)
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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

−18.732498353915750575124660336157, −17.82181899211921104002852556244, −17.010802082805945071578050217564, −16.78267969792265340768529503289, −15.63385107143008800309511912942, −15.57449941313016576471668724185, −14.71042469462444914858157976289, −13.76880530002070155349447344901, −13.35304045979976916847950686034, −12.33783118564411799130138125405, −11.555521686635049512035206894908, −10.91400525596010029370360053778, −10.58763604202314711093262251942, −9.5701372788945266373625958896, −8.88784822337978129974893202348, −8.504962594436457437139216488596, −7.34330029331213051107016350231, −6.46763595414252627512545243300, −5.93254904890594121470999928446, −4.99365279755422367659999725560, −4.42699599595462620476514237944, −3.69968279859176312240225291151, −2.801310953454575702457442656101, −1.99510366537261966560110106459, −0.61563215853184705882698248201, 0.09498287590206036728326693167, 1.09449382965988804883175255941, 1.96616551601149157771719891714, 2.66439735368248405522821992094, 3.54553578902025189155698765674, 4.80038079039970284311340443007, 5.222245534473784119659093316446, 6.128068484983783321380381065852, 6.904262633123029305713816102797, 7.34424403046147893501431848683, 8.357453881695214164463903299563, 8.66099493061092842292708730503, 9.85961663210965798584989811631, 10.76067391788862678725045816357, 10.95531739975917818078884746952, 12.08536214169544034918986928957, 12.71533661330852518834195220104, 13.02731555095016540131965097103, 13.79486149987951899389636408531, 14.70986106398678736396319195304, 15.265688713572561062429317321973, 16.10322102626254843655070278363, 16.82900861837229267009514063941, 17.60979782468399822480984269448, 18.04749123467354509733435339170

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