Properties

Label 1-47e2-2209.1033-r1-0-0
Degree $1$
Conductor $2209$
Sign $-0.323 - 0.946i$
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  + (−0.824 + 0.565i)2-s + (0.593 − 0.805i)3-s + (0.359 − 0.933i)4-s + (0.979 − 0.199i)5-s + (−0.0334 + 0.999i)6-s + (−0.420 + 0.907i)7-s + (0.231 + 0.972i)8-s + (−0.296 − 0.955i)9-s + (−0.695 + 0.718i)10-s + (0.944 + 0.328i)11-s + (−0.538 − 0.842i)12-s + (0.944 + 0.328i)13-s + (−0.166 − 0.986i)14-s + (0.420 − 0.907i)15-s + (−0.741 − 0.670i)16-s + (−0.892 − 0.451i)17-s + ⋯
L(s)  = 1  + (−0.824 + 0.565i)2-s + (0.593 − 0.805i)3-s + (0.359 − 0.933i)4-s + (0.979 − 0.199i)5-s + (−0.0334 + 0.999i)6-s + (−0.420 + 0.907i)7-s + (0.231 + 0.972i)8-s + (−0.296 − 0.955i)9-s + (−0.695 + 0.718i)10-s + (0.944 + 0.328i)11-s + (−0.538 − 0.842i)12-s + (0.944 + 0.328i)13-s + (−0.166 − 0.986i)14-s + (0.420 − 0.907i)15-s + (−0.741 − 0.670i)16-s + (−0.892 − 0.451i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2209\)    =    \(47^{2}\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(237.390\)
Root analytic conductor: \(237.390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2209} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2209,\ (1:\ ),\ -0.323 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056208029 - 1.478141280i\)
\(L(\frac12)\) \(\approx\) \(1.056208029 - 1.478141280i\)
\(L(1)\) \(\approx\) \(1.031325931 - 0.1722319168i\)
\(L(1)\) \(\approx\) \(1.031325931 - 0.1722319168i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (-0.824 + 0.565i)T \)
3 \( 1 + (0.593 - 0.805i)T \)
5 \( 1 + (0.979 - 0.199i)T \)
7 \( 1 + (-0.420 + 0.907i)T \)
11 \( 1 + (0.944 + 0.328i)T \)
13 \( 1 + (0.944 + 0.328i)T \)
17 \( 1 + (-0.892 - 0.451i)T \)
19 \( 1 + (0.979 - 0.199i)T \)
23 \( 1 + (-0.695 - 0.718i)T \)
29 \( 1 + (-0.359 - 0.933i)T \)
31 \( 1 + (-0.231 - 0.972i)T \)
37 \( 1 + (-0.892 + 0.451i)T \)
41 \( 1 + (0.997 + 0.0667i)T \)
43 \( 1 + (-0.359 - 0.933i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.979 - 0.199i)T \)
61 \( 1 + (0.964 + 0.264i)T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + (0.0334 + 0.999i)T \)
79 \( 1 + (-0.166 - 0.986i)T \)
83 \( 1 + (-0.979 + 0.199i)T \)
89 \( 1 + (-0.420 - 0.907i)T \)
97 \( 1 + (0.480 - 0.876i)T \)
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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

−19.85563516850730352353043321108, −19.32515577682642686446119497694, −18.197374915851040751967767125213, −17.70159198546258101832343145441, −16.905930032043003662430415390910, −16.26879316048935137066271437413, −15.763135141690992463783977337, −14.58935741842411796859133274595, −13.78249150749198543556476848209, −13.44423881707695332145667224460, −12.49496621451233828845815185468, −11.24431413936666166666473410909, −10.77944130584824340831448038703, −10.17017798139951594612229524630, −9.39627604576076287960521073069, −9.007902625956964955304362309741, −8.158033989196244564713350895351, −7.18229156439138956355185625675, −6.43540448192998476996392702381, −5.4498720871000439332024248263, −4.1298137433295092989646065692, −3.561730539875833347163045946850, −2.9389150687224424113980915739, −1.74135024337380011911116674149, −1.15808600430724136129273577409, 0.35705197886624995700575686821, 1.36862967531699024915404593837, 2.07304680787786993957073919604, 2.6931843174374656064209509169, 4.064079796249561321502690296717, 5.393198739832433891673406262620, 6.14157135497788158861935185868, 6.54127720386039115448505771640, 7.32393358476334526047859376277, 8.41855341288060283793568197511, 8.962006894326342128820624239760, 9.38666128330324210579076231272, 10.08210216408168334722898297117, 11.39224333428736192859612977033, 11.92160908399930264272247098751, 12.958546175879837551566142051967, 13.728892156029826768872770075161, 14.16923594218754698960297915025, 15.07633127773407235527013110991, 15.71904510014605771625046855432, 16.56182442688261536851483221339, 17.381565076246239417824139210264, 17.99599178348764796078651776309, 18.51939783584509426492960971113, 19.03297023396629349340599062009

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