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Properties

Label 1-380-380.279-r0-0-0
Degree $1$
Conductor $380$
Sign $-0.766 - 0.642i$
Analytic cond. $1.76471$
Root an. cond. $1.76471$
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  + (−0.766 − 0.642i)3-s + (−0.5 − 0.866i)7-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (0.766 − 0.642i)13-s + (−0.173 + 0.984i)17-s + (−0.173 + 0.984i)21-s + (−0.939 − 0.342i)23-s + (0.5 − 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.939 + 0.342i)33-s + 37-s − 39-s + (−0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)3-s + (−0.5 − 0.866i)7-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (0.766 − 0.642i)13-s + (−0.173 + 0.984i)17-s + (−0.173 + 0.984i)21-s + (−0.939 − 0.342i)23-s + (0.5 − 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.939 + 0.342i)33-s + 37-s − 39-s + (−0.766 − 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(1.76471\)
Root analytic conductor: \(1.76471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 380,\ (0:\ ),\ -0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2403998985 - 0.6611645770i\)
\(L(\frac12)\) \(\approx\) \(0.2403998985 - 0.6611645770i\)
\(L(1)\) \(\approx\) \(0.6690191050 - 0.3374014621i\)
\(L(1)\) \(\approx\) \(0.6690191050 - 0.3374014621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−25.06036397416700602988240419881, −23.74825145577925316207417772797, −23.09945797646864687520823229040, −22.12376339077905672794795645030, −21.719553577748361925967306881357, −20.590977403386911826518036257073, −19.792107373857319101367786868841, −18.35020954314763946900591947342, −18.06780299882061483356003897958, −16.734788195695663873300416442409, −16.09943269709647623047891285779, −15.32505277778337721077671797497, −14.39839891382913466316593233704, −13.10354463519421651351321338759, −12.006463979489611157744765548478, −11.585929880949926392390670947428, −10.32915729186263560687802305374, −9.439811135400947310659360655623, −8.78965375432200438790750531522, −7.082161192698138522348349363695, −6.272335611309063624990622294318, −5.28671024760091134001978757920, −4.289757953258531599554723720069, −3.182118885279409964206867168148, −1.62626292385108545995361178516, 0.47796156144797367238237325839, 1.71329963435592205239185473823, 3.37519941306891893207696962489, 4.39555836888757039887139197039, 6.02008801602111723774541709747, 6.24897663580895002486747082315, 7.59449925093027722951904406571, 8.37740604284287436893312989609, 9.86593055177610813839423108636, 10.795920212624770943908739668017, 11.4588661336175032796391813398, 12.676430933718059530936545347108, 13.3299417922047919250181109549, 14.11994969390688850022974905220, 15.52892033897539792929867008084, 16.53140585393603014936493249642, 17.05454502313103055914579917697, 18.02089705145039343104112392515, 18.94264664804494566394218943312, 19.67646770757254872968871362056, 20.62283658003018014749524478892, 21.93902974492573390201796860051, 22.47870922649275610979297906244, 23.48346549151215519881347026860, 23.98673894817013590334807457827

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