Properties

Label 1-47e2-2209.1025-r1-0-0
Degree $1$
Conductor $2209$
Sign $-0.652 - 0.757i$
Analytic cond. $237.390$
Root an. cond. $237.390$
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  + (−0.958 − 0.283i)2-s + (0.751 + 0.659i)3-s + (0.838 + 0.544i)4-s + (−0.495 − 0.868i)5-s + (−0.533 − 0.846i)6-s + (0.574 − 0.818i)7-s + (−0.650 − 0.759i)8-s + (0.128 + 0.991i)9-s + (0.228 + 0.973i)10-s + (0.0450 − 0.998i)11-s + (0.271 + 0.962i)12-s + (0.0914 + 0.995i)13-s + (−0.782 + 0.622i)14-s + (0.200 − 0.979i)15-s + (0.407 + 0.913i)16-s + (−0.948 − 0.317i)17-s + ⋯
L(s)  = 1  + (−0.958 − 0.283i)2-s + (0.751 + 0.659i)3-s + (0.838 + 0.544i)4-s + (−0.495 − 0.868i)5-s + (−0.533 − 0.846i)6-s + (0.574 − 0.818i)7-s + (−0.650 − 0.759i)8-s + (0.128 + 0.991i)9-s + (0.228 + 0.973i)10-s + (0.0450 − 0.998i)11-s + (0.271 + 0.962i)12-s + (0.0914 + 0.995i)13-s + (−0.782 + 0.622i)14-s + (0.200 − 0.979i)15-s + (0.407 + 0.913i)16-s + (−0.948 − 0.317i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2209\)    =    \(47^{2}\)
Sign: $-0.652 - 0.757i$
Analytic conductor: \(237.390\)
Root analytic conductor: \(237.390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2209} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2209,\ (1:\ ),\ -0.652 - 0.757i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5557296165 - 1.212307965i\)
\(L(\frac12)\) \(\approx\) \(0.5557296165 - 1.212307965i\)
\(L(1)\) \(\approx\) \(0.8423320708 - 0.2152238744i\)
\(L(1)\) \(\approx\) \(0.8423320708 - 0.2152238744i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (-0.958 - 0.283i)T \)
3 \( 1 + (0.751 + 0.659i)T \)
5 \( 1 + (-0.495 - 0.868i)T \)
7 \( 1 + (0.574 - 0.818i)T \)
11 \( 1 + (0.0450 - 0.998i)T \)
13 \( 1 + (0.0914 + 0.995i)T \)
17 \( 1 + (-0.948 - 0.317i)T \)
19 \( 1 + (0.900 - 0.435i)T \)
23 \( 1 + (-0.555 - 0.831i)T \)
29 \( 1 + (0.600 + 0.799i)T \)
31 \( 1 + (0.245 - 0.969i)T \)
37 \( 1 + (0.743 - 0.668i)T \)
41 \( 1 + (0.172 - 0.985i)T \)
43 \( 1 + (0.994 + 0.107i)T \)
53 \( 1 + (0.962 + 0.269i)T \)
59 \( 1 + (-0.800 + 0.599i)T \)
61 \( 1 + (-0.858 + 0.512i)T \)
67 \( 1 + (-0.460 - 0.887i)T \)
71 \( 1 + (-0.576 + 0.816i)T \)
73 \( 1 + (0.807 + 0.589i)T \)
79 \( 1 + (0.449 + 0.893i)T \)
83 \( 1 + (-0.749 + 0.662i)T \)
89 \( 1 + (0.579 - 0.815i)T \)
97 \( 1 + (-0.604 - 0.796i)T \)
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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

−19.76064594881696098489460749857, −19.01601451150326526211766027387, −18.14666024972508709872161057443, −17.96770327649765379633643258332, −17.425373233201025721587607039740, −15.86041864131479414695176323238, −15.45478404826763884574304014509, −14.918354915583389565482145704365, −14.333280862940467135807350865172, −13.37999103020043237451254298248, −12.1626661948299211328983785946, −11.88449924695442245493377679176, −10.92872931656504629770027075482, −10.063480911689875925359485384141, −9.393616725749484116640621749481, −8.499590099599191950182369175, −7.77841038950328834488802417325, −7.54440647546565234437227673705, −6.491799492898138974286666927492, −5.94315253005783627854897961942, −4.697052223175076526192311985063, −3.380146072176507074462479114726, −2.61201609521335189041441013292, −1.969867809675138484981122192, −1.035689013921352483639567911783, 0.31500971510432101295982986376, 1.12125091332552042410559737789, 2.130376345592780872910317308323, 3.05969533779764854481443344980, 4.16577100920516347582342696092, 4.35730120048231785173653998940, 5.67051106109840392454703347927, 6.962854724935885331476027252010, 7.63582181071254139301892871704, 8.36897711796740876714541070628, 8.9408437463093542585515066994, 9.417858927770345731074283506648, 10.45600572535496210546234258877, 11.14050768763929076317673298610, 11.596079706539361517074010540740, 12.64524713648753360942957475577, 13.67978614599950195980001264541, 14.05472979340992294854522412847, 15.23069683360987640291768700614, 15.935178960222874484476713997720, 16.48802998613621583890404222832, 16.85811421225892563774650709946, 17.90867013833128563464240118590, 18.68178777179112960443668061811, 19.60503277295037385085203320266

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