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Properties

Label 1-47e2-2209.189-r0-0-0
Degree $1$
Conductor $2209$
Sign $-0.998 + 0.0554i$
Analytic cond. $10.2585$
Root an. cond. $10.2585$
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  + (−0.0334 − 0.999i)2-s + (0.920 − 0.390i)3-s + (−0.997 + 0.0667i)4-s + (0.860 − 0.509i)5-s + (−0.420 − 0.907i)6-s + (0.593 − 0.805i)7-s + (0.100 + 0.994i)8-s + (0.695 − 0.718i)9-s + (−0.538 − 0.842i)10-s + (0.359 − 0.933i)11-s + (−0.892 + 0.451i)12-s + (0.359 − 0.933i)13-s + (−0.824 − 0.565i)14-s + (0.593 − 0.805i)15-s + (0.991 − 0.133i)16-s + (−0.979 − 0.199i)17-s + ⋯
L(s)  = 1  + (−0.0334 − 0.999i)2-s + (0.920 − 0.390i)3-s + (−0.997 + 0.0667i)4-s + (0.860 − 0.509i)5-s + (−0.420 − 0.907i)6-s + (0.593 − 0.805i)7-s + (0.100 + 0.994i)8-s + (0.695 − 0.718i)9-s + (−0.538 − 0.842i)10-s + (0.359 − 0.933i)11-s + (−0.892 + 0.451i)12-s + (0.359 − 0.933i)13-s + (−0.824 − 0.565i)14-s + (0.593 − 0.805i)15-s + (0.991 − 0.133i)16-s + (−0.979 − 0.199i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2209\)    =    \(47^{2}\)
Sign: $-0.998 + 0.0554i$
Analytic conductor: \(10.2585\)
Root analytic conductor: \(10.2585\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2209} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2209,\ (0:\ ),\ -0.998 + 0.0554i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07369166371 - 2.656551478i\)
\(L(\frac12)\) \(\approx\) \(-0.07369166371 - 2.656551478i\)
\(L(1)\) \(\approx\) \(0.9411804741 - 1.316150136i\)
\(L(1)\) \(\approx\) \(0.9411804741 - 1.316150136i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (-0.0334 - 0.999i)T \)
3 \( 1 + (0.920 - 0.390i)T \)
5 \( 1 + (0.860 - 0.509i)T \)
7 \( 1 + (0.593 - 0.805i)T \)
11 \( 1 + (0.359 - 0.933i)T \)
13 \( 1 + (0.359 - 0.933i)T \)
17 \( 1 + (-0.979 - 0.199i)T \)
19 \( 1 + (0.860 - 0.509i)T \)
23 \( 1 + (-0.538 + 0.842i)T \)
29 \( 1 + (-0.997 - 0.0667i)T \)
31 \( 1 + (0.100 + 0.994i)T \)
37 \( 1 + (-0.979 + 0.199i)T \)
41 \( 1 + (-0.645 + 0.763i)T \)
43 \( 1 + (-0.997 - 0.0667i)T \)
53 \( 1 + T \)
59 \( 1 + (0.860 + 0.509i)T \)
61 \( 1 + (-0.944 + 0.328i)T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + (-0.420 + 0.907i)T \)
79 \( 1 + (-0.824 - 0.565i)T \)
83 \( 1 + (0.860 - 0.509i)T \)
89 \( 1 + (0.593 + 0.805i)T \)
97 \( 1 + (0.231 + 0.972i)T \)
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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

−20.221915087020647679929733042823, −19.065623414641825786575008614043, −18.47186610274971209233062024590, −18.05456513454375375711764914342, −17.14613980792906652569312215469, −16.45027615936546606766838408953, −15.44531009210317957783067690101, −15.10130126550405962143583759886, −14.341247191316637908355355500328, −13.943311943619833240022511420851, −13.19329138084173542093429865797, −12.290505442723856591138609406502, −11.18957000662302043277429929282, −10.12146249224071266906297891173, −9.578653968478320961244578045133, −8.920973653130108897366857073905, −8.37257325580390346534685764198, −7.34589541187326069077579389800, −6.75271900777812033941616001118, −5.85975813743333308668858313597, −5.02115994171601905923951253061, −4.273334769230186850601988042870, −3.438250993902439715707189749770, −2.04782289573193281324249258352, −1.82053031105942976749883790343, 0.79754909537915168964638785669, 1.40623587025682894763507921474, 2.18735901601419820638648446067, 3.227705103053978866391060128114, 3.75484260926847895243316457993, 4.837411384928597446677918610173, 5.54268294063630590058935778036, 6.70445918912544740078200656920, 7.723459341878917897658153511426, 8.51337198288782690250909303689, 8.94364960535902542215207125188, 9.81153844255606544131588487590, 10.440287804705320449501646190425, 11.32287469703647310850266144898, 12.04596271005911354936490096272, 13.22786674734390776135460921660, 13.38683555355000702528367283754, 13.897967055394940582457521099216, 14.64231871942561948381623869187, 15.671921037500509713471802190144, 16.71900573142753961557229214450, 17.638614859252139008666003589758, 17.91731540015423435302206635535, 18.67701907488238154652982304426, 19.72705754463896244446509509653

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