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Properties

Label 1-385-385.54-r0-0-0
Degree $1$
Conductor $385$
Sign $0.386 - 0.922i$
Analytic cond. $1.78793$
Root an. cond. $1.78793$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 6-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)12-s − 13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)24-s + (0.5 + 0.866i)26-s + 27-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 6-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)12-s − 13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)24-s + (0.5 + 0.866i)26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5254205034 - 0.3495002035i\)
\(L(\frac12)\) \(\approx\) \(0.5254205034 - 0.3495002035i\)
\(L(1)\) \(\approx\) \(0.6215715538 - 0.1439075860i\)
\(L(1)\) \(\approx\) \(0.6215715538 - 0.1439075860i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + T \)
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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

−24.58797133111151527928637811091, −24.05418513079002170776839049162, −23.04735554882038583843844679450, −22.55996999073485612648416250404, −21.30821801565187068211530470894, −19.82846254110980766931980435159, −19.158794182393583543075992027887, −18.46839513825478533614541371193, −17.472333015714335987493893639, −16.91902265931936550111392873883, −16.13482526584322596551422704214, −14.75642513521383202907484204962, −14.28696536846224388110255710110, −12.97521823561866168059010127821, −12.33598243811417315463659886239, −10.95838109945473469875691261631, −10.17902514328785465339646567749, −8.932107665638851576817139472623, −7.91978786876606235362117143584, −7.25136138892645127792608941030, −6.21523292164387260914180129515, −5.49496648358563867061985435895, −4.31695079820876490082965404202, −2.332116534347328118092806870875, −1.08082390451834776784417667007, 0.57920505104986934551074883326, 2.367770147281430974349345359084, 3.418603526023900104249346047649, 4.53105565348502593031276080389, 5.34106665826352182358197455111, 6.92347842814365402513362755074, 8.0425533109106350307505813106, 9.48661551191362547921272010277, 9.57669705121890844341822562839, 10.88441529159120179094119306048, 11.491014236244937599193477322328, 12.367190200684418112450737803086, 13.42895508228190152436546840925, 14.64291976510040136034956859090, 15.62946384593915943567701642862, 16.74532540504889155565937483853, 17.23549134621272942627168156917, 18.161876721262896463512595438861, 19.21120467106321706439144926546, 20.06474682359719859308564471002, 20.92956979691966513569059239040, 21.60453583798372944637638036179, 22.40158014148804999068941551165, 23.067303981259930711449148448830, 24.285464368180448177521182804346

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