Properties

Label 1-47e2-2209.986-r1-0-0
Degree $1$
Conductor $2209$
Sign $0.625 - 0.780i$
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  + (−0.538 + 0.842i)2-s + (0.860 + 0.509i)3-s + (−0.420 − 0.907i)4-s + (0.944 − 0.328i)5-s + (−0.892 + 0.451i)6-s + (−0.979 − 0.199i)7-s + (0.991 + 0.133i)8-s + (0.480 + 0.876i)9-s + (−0.231 + 0.972i)10-s + (0.0334 − 0.999i)11-s + (0.100 − 0.994i)12-s + (0.0334 − 0.999i)13-s + (0.695 − 0.718i)14-s + (0.979 + 0.199i)15-s + (−0.645 + 0.763i)16-s + (0.964 − 0.264i)17-s + ⋯
L(s)  = 1  + (−0.538 + 0.842i)2-s + (0.860 + 0.509i)3-s + (−0.420 − 0.907i)4-s + (0.944 − 0.328i)5-s + (−0.892 + 0.451i)6-s + (−0.979 − 0.199i)7-s + (0.991 + 0.133i)8-s + (0.480 + 0.876i)9-s + (−0.231 + 0.972i)10-s + (0.0334 − 0.999i)11-s + (0.100 − 0.994i)12-s + (0.0334 − 0.999i)13-s + (0.695 − 0.718i)14-s + (0.979 + 0.199i)15-s + (−0.645 + 0.763i)16-s + (0.964 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2209\)    =    \(47^{2}\)
Sign: $0.625 - 0.780i$
Analytic conductor: \(237.390\)
Root analytic conductor: \(237.390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2209} (986, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2209,\ (1:\ ),\ 0.625 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.095461440 - 1.006107066i\)
\(L(\frac12)\) \(\approx\) \(2.095461440 - 1.006107066i\)
\(L(1)\) \(\approx\) \(1.180757164 + 0.1906138311i\)
\(L(1)\) \(\approx\) \(1.180757164 + 0.1906138311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (-0.538 + 0.842i)T \)
3 \( 1 + (0.860 + 0.509i)T \)
5 \( 1 + (0.944 - 0.328i)T \)
7 \( 1 + (-0.979 - 0.199i)T \)
11 \( 1 + (0.0334 - 0.999i)T \)
13 \( 1 + (0.0334 - 0.999i)T \)
17 \( 1 + (0.964 - 0.264i)T \)
19 \( 1 + (0.944 - 0.328i)T \)
23 \( 1 + (-0.231 - 0.972i)T \)
29 \( 1 + (0.420 - 0.907i)T \)
31 \( 1 + (-0.991 - 0.133i)T \)
37 \( 1 + (0.964 + 0.264i)T \)
41 \( 1 + (-0.593 + 0.805i)T \)
43 \( 1 + (0.420 - 0.907i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.944 - 0.328i)T \)
61 \( 1 + (-0.824 + 0.565i)T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + (0.892 + 0.451i)T \)
79 \( 1 + (0.695 - 0.718i)T \)
83 \( 1 + (-0.944 + 0.328i)T \)
89 \( 1 + (-0.979 + 0.199i)T \)
97 \( 1 + (-0.741 + 0.670i)T \)
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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

−19.84744838774615634830949675409, −18.85186922371602389934133637168, −18.36880580683206129740820181006, −17.899024497072668383847784272705, −16.90725554884470584620478878295, −16.2768851207528277280844553678, −15.20313077474238060652469828963, −14.21102998106904082817485184420, −13.83133258927323277287812486405, −12.95411785735571883072887750549, −12.4412881205984371930391018446, −11.80226615721195489865145114041, −10.58963566612242672632979985214, −9.77053611977996048135328943812, −9.46353996066075818776353945763, −8.93094855177574362477417919199, −7.69142747263593000776814915975, −7.18125925976041847016011629710, −6.38695792200426627503246010508, −5.29372281440210891362715839743, −3.973254896846371502796674437201, −3.29870100779899069028530191476, −2.572883338594030787806430625756, −1.74513618349760295493638529960, −1.20331066245108230061203315637, 0.42970402264156116970703609067, 1.20170966553327929766063344095, 2.55671376251586719495612228018, 3.22947630198727234468296365329, 4.31457956544833909180448033959, 5.392959399897316876077224560567, 5.81128838438686869994323406731, 6.74276711461106814452265460471, 7.71353047166860677541103381890, 8.34328845062458903745553348575, 9.14569836865162246396246793545, 9.70213134852815753248032876416, 10.21295243990914925216262726284, 10.90193146813440620555753258533, 12.43483213707485136022032856863, 13.41955964196423447714945560274, 13.64955308333896458901038145090, 14.40026979684630113169994574853, 15.22384715999409646527001525241, 15.95296373376920014337257529031, 16.63014508829124134800621022085, 16.86666794471182544295878792025, 18.205078831424990450213557687165, 18.518593364880174576905502275938, 19.46385825864188178371125560961

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