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Properties

Label 1-3381-3381.26-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.855 + 0.518i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (−0.990 + 0.135i)2-s + (0.963 − 0.268i)4-s + (0.976 − 0.215i)5-s + (−0.917 + 0.396i)8-s + (−0.938 + 0.346i)10-s + (0.0882 + 0.996i)11-s + (−0.970 − 0.242i)13-s + (0.855 − 0.517i)16-s + (−0.248 − 0.968i)17-s + (0.888 + 0.458i)19-s + (0.882 − 0.470i)20-s + (−0.222 − 0.974i)22-s + (0.906 − 0.421i)25-s + (0.994 + 0.108i)26-s + (0.714 − 0.699i)29-s + ⋯
L(s)  = 1  + (−0.990 + 0.135i)2-s + (0.963 − 0.268i)4-s + (0.976 − 0.215i)5-s + (−0.917 + 0.396i)8-s + (−0.938 + 0.346i)10-s + (0.0882 + 0.996i)11-s + (−0.970 − 0.242i)13-s + (0.855 − 0.517i)16-s + (−0.248 − 0.968i)17-s + (0.888 + 0.458i)19-s + (0.882 − 0.470i)20-s + (−0.222 − 0.974i)22-s + (0.906 − 0.421i)25-s + (0.994 + 0.108i)26-s + (0.714 − 0.699i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.855 + 0.518i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.855 + 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.242443886 + 0.3472751201i\)
\(L(\frac12)\) \(\approx\) \(1.242443886 + 0.3472751201i\)
\(L(1)\) \(\approx\) \(0.8504633674 + 0.07544710825i\)
\(L(1)\) \(\approx\) \(0.8504633674 + 0.07544710825i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.990 + 0.135i)T \)
5 \( 1 + (0.976 - 0.215i)T \)
11 \( 1 + (0.0882 + 0.996i)T \)
13 \( 1 + (-0.970 - 0.242i)T \)
17 \( 1 + (-0.248 - 0.968i)T \)
19 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (0.714 - 0.699i)T \)
31 \( 1 + (0.327 + 0.945i)T \)
37 \( 1 + (0.155 + 0.987i)T \)
41 \( 1 + (-0.301 + 0.953i)T \)
43 \( 1 + (0.917 + 0.396i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (-0.665 + 0.746i)T \)
59 \( 1 + (0.938 - 0.346i)T \)
61 \( 1 + (0.403 - 0.915i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (-0.488 - 0.872i)T \)
73 \( 1 + (0.751 - 0.659i)T \)
79 \( 1 + (0.580 - 0.814i)T \)
83 \( 1 + (-0.862 + 0.505i)T \)
89 \( 1 + (-0.942 + 0.333i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−18.81977136012728137085140493102, −17.87593726306522684263987282430, −17.55110261202285293520528954702, −16.83994880736027231784452376739, −16.24267707303472192297953210396, −15.467983234446430340742953292610, −14.56345106976976459717665730698, −14.04690463584088320456977902636, −13.07924562499314798846632220942, −12.45522553752807120011178620045, −11.47620028261050879801705674992, −10.992075899640228622195089377721, −10.12009293653573840241998894317, −9.70869852489150888155955197833, −8.862513810524050960303247921634, −8.36633992958806338575304039961, −7.2927170854031979718047864449, −6.789574377424590860978185095074, −5.89163605849631496224887034500, −5.382947483765122875825939644983, −4.068078104376885887488448440353, −3.03344783755967532241913219194, −2.43085339690346540071159139729, −1.60467028563112386903536453046, −0.6522063064115382048860950389, 0.86079541939337393365178065441, 1.705203009277863763721427340673, 2.48617780044798234398162565139, 3.13155599933200848583363282518, 4.78481424610614475205901432134, 5.110840244666382836352658847296, 6.23599174690805988202757091116, 6.76610664192448374072781467672, 7.57910028179412500003864486255, 8.21576810854098676653538904431, 9.26398432785507324373721702008, 9.73204732599275973900875959866, 10.05363084321084438788583739695, 10.9915745268183996304724603101, 11.97656244489412597650783778629, 12.332284394729311109832611216336, 13.360950221540765107890837403915, 14.19614395850451079543322660490, 14.76181747736357653669642180794, 15.61503991317901724956996986984, 16.25347760070146038264405125016, 17.01460713290625624436956539943, 17.59503513957033714757322469573, 18.00780519217916916570786770203, 18.66468876369122017933728715949

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