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Properties

Label 1-385-385.24-r0-0-0
Degree $1$
Conductor $385$
Sign $0.500 - 0.866i$
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  + (0.913 − 0.406i)2-s + (−0.978 + 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.5 + 0.866i)12-s + (0.809 + 0.587i)13-s + (−0.104 − 0.994i)16-s + (−0.913 − 0.406i)17-s + (0.669 − 0.743i)18-s + (0.669 + 0.743i)19-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)24-s + (0.978 + 0.207i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (−0.978 + 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.5 + 0.866i)12-s + (0.809 + 0.587i)13-s + (−0.104 − 0.994i)16-s + (−0.913 − 0.406i)17-s + (0.669 − 0.743i)18-s + (0.669 + 0.743i)19-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)24-s + (0.978 + 0.207i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.500 - 0.866i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.500 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.491645006 - 0.8611569499i\)
\(L(\frac12)\) \(\approx\) \(1.491645006 - 0.8611569499i\)
\(L(1)\) \(\approx\) \(1.341652196 - 0.4233012787i\)
\(L(1)\) \(\approx\) \(1.341652196 - 0.4233012787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.978 + 0.207i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−24.478120989801055539690570625367, −23.5481839425748898586189415156, −23.11808483414950597352895753482, −22.04840731269392831529100357663, −21.65374229126376593521494504144, −20.479878988493557693840379631082, −19.55876219836029911863205462655, −18.09534383450268256448830518194, −17.59728113526462076940212993843, −16.53971851540447217554346800070, −15.78598388490911277829020553493, −15.08557147466683702030163455331, −13.69127054335312677713875238024, −13.090944012739873150164416378731, −12.23697898143479546617661175249, −11.19336595185454439205939082735, −10.71695577853355602868067988824, −9.05883115853979496912131351180, −7.74201692531398608408535146642, −6.88732445964033640046287567723, −5.981838115940725461767086136619, −5.17846324079470967092510785924, −4.20119450781181683694434445277, −2.964422849193436602494942453408, −1.41032174467077699712138390995, 0.99623572733010675116548814123, 2.368819177577806356202615110105, 3.8899547865142244886753748758, 4.54435423971188197061862819928, 5.7304806356735187496790010293, 6.3716874026209525403686798585, 7.42814237597683048793383708456, 9.18254026209124965603683007446, 10.172478617979560473805385148982, 11.16090283960288177330310150562, 11.63085108150359873559817602285, 12.68202473538537847584857019406, 13.46945076275209106680700696605, 14.504169750604194069039195295509, 15.61443177265754866867207210184, 16.17432436222701253543255223940, 17.180994325411159969499821779658, 18.415373160470717489158806863960, 19.008268627098591424911463013284, 20.47273907542472021877677525346, 20.90433946192606067883663890450, 21.96249540893236875475650703101, 22.62386605505421823064743861420, 23.22270859064270210482167942303, 24.21086436781101262070865000254

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