Properties

Label 1-1375-1375.138-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.952 + 0.306i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (−0.248 + 0.968i)2-s + (0.904 − 0.425i)3-s + (−0.876 − 0.481i)4-s + (0.187 + 0.982i)6-s + (−0.951 + 0.309i)7-s + (0.684 − 0.728i)8-s + (0.637 − 0.770i)9-s + (−0.998 − 0.0627i)12-s + (0.770 + 0.637i)13-s + (−0.0627 − 0.998i)14-s + (0.535 + 0.844i)16-s + (−0.904 − 0.425i)17-s + (0.587 + 0.809i)18-s + (−0.187 − 0.982i)19-s + (−0.728 + 0.684i)21-s + ⋯
L(s)  = 1  + (−0.248 + 0.968i)2-s + (0.904 − 0.425i)3-s + (−0.876 − 0.481i)4-s + (0.187 + 0.982i)6-s + (−0.951 + 0.309i)7-s + (0.684 − 0.728i)8-s + (0.637 − 0.770i)9-s + (−0.998 − 0.0627i)12-s + (0.770 + 0.637i)13-s + (−0.0627 − 0.998i)14-s + (0.535 + 0.844i)16-s + (−0.904 − 0.425i)17-s + (0.587 + 0.809i)18-s + (−0.187 − 0.982i)19-s + (−0.728 + 0.684i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.952 + 0.306i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.952 + 0.306i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.517837312 + 0.2379541596i\)
\(L(\frac12)\) \(\approx\) \(1.517837312 + 0.2379541596i\)
\(L(1)\) \(\approx\) \(1.079004793 + 0.2611045943i\)
\(L(1)\) \(\approx\) \(1.079004793 + 0.2611045943i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.248 + 0.968i)T \)
3 \( 1 + (0.904 - 0.425i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (0.770 + 0.637i)T \)
17 \( 1 + (-0.904 - 0.425i)T \)
19 \( 1 + (-0.187 - 0.982i)T \)
23 \( 1 + (0.844 + 0.535i)T \)
29 \( 1 + (-0.992 + 0.125i)T \)
31 \( 1 + (0.728 + 0.684i)T \)
37 \( 1 + (0.770 + 0.637i)T \)
41 \( 1 + (0.637 - 0.770i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (0.982 + 0.187i)T \)
53 \( 1 + (-0.982 - 0.187i)T \)
59 \( 1 + (0.929 + 0.368i)T \)
61 \( 1 + (0.929 - 0.368i)T \)
67 \( 1 + (0.904 + 0.425i)T \)
71 \( 1 + (-0.425 - 0.904i)T \)
73 \( 1 + (0.248 - 0.968i)T \)
79 \( 1 + (0.728 - 0.684i)T \)
83 \( 1 + (0.684 - 0.728i)T \)
89 \( 1 + (-0.0627 - 0.998i)T \)
97 \( 1 + (0.481 - 0.876i)T \)
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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

−20.47199926891279738880913488560, −20.29981430855174165368431083896, −19.29506539907257602055105792683, −18.90085143173438154509506559411, −18.064516966240683406656901083228, −16.978945785478182588357202870624, −16.34867097355128988209110468189, −15.422194735611007510041957000868, −14.61847488607679783925619495273, −13.68063804720127813860993336207, −12.998846113651742948428965258241, −12.69968482888166692128621702065, −11.252920422060871657994138584966, −10.669197076946834279395218864495, −9.8778922382210857228035796186, −9.321006240526871937674007006220, −8.43581221288603509864052053622, −7.86843635182794885014440541481, −6.69217964004160853471937025117, −5.52469672580711735209701602433, −4.23821493748357940411994482664, −3.79975095160601392999299666254, −2.9288058064389781947272120613, −2.16956516383783353572033553938, −0.95107872252985878680475309941, 0.74295124646680660838147577774, 2.00627547072671815523297576096, 3.12339668662398742746510628523, 3.99207851606982883763717004422, 4.98822663961965886089152969549, 6.21128954918924030587995204218, 6.75231549726243802370912131037, 7.37547903108524638776123661448, 8.469690650856592315654002424031, 9.12219768676685503416464894678, 9.43152433262320675291585785731, 10.59484434914373388224268098906, 11.72774646915273325694535013964, 13.027494957128013717211955637146, 13.24961443196894581731347243448, 13.99760056389046002484142699995, 14.94154942887184445770795931095, 15.606155153872234065673475470611, 16.05196358319123375022805689663, 17.10872012812904752406997298709, 17.9108045236229840829456493234, 18.69359637672352548764545486013, 19.17736730877091109631717720418, 19.83907772251919145768579577056, 20.76472342639111438608347782338

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