Properties

Label 1-385-385.4-r0-0-0
Degree $1$
Conductor $385$
Sign $0.716 - 0.698i$
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  + (0.978 + 0.207i)2-s + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (0.309 − 0.951i)6-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (0.5 − 0.866i)12-s + (−0.309 − 0.951i)13-s + (0.669 + 0.743i)16-s + (0.978 − 0.207i)17-s + (−0.913 − 0.406i)18-s + (0.913 − 0.406i)19-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)24-s + (−0.104 − 0.994i)26-s + (−0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (0.309 − 0.951i)6-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (0.5 − 0.866i)12-s + (−0.309 − 0.951i)13-s + (0.669 + 0.743i)16-s + (0.978 − 0.207i)17-s + (−0.913 − 0.406i)18-s + (0.913 − 0.406i)19-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)24-s + (−0.104 − 0.994i)26-s + (−0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.716 - 0.698i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.716 - 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.309089588 - 0.9392437677i\)
\(L(\frac12)\) \(\approx\) \(2.309089588 - 0.9392437677i\)
\(L(1)\) \(\approx\) \(1.881030745 - 0.4182945043i\)
\(L(1)\) \(\approx\) \(1.881030745 - 0.4182945043i\)

Euler product

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

−24.56824082062389581813862345062, −23.40731557731552501555334941511, −22.86866963085212395141887033348, −21.80086039426666165755592733455, −21.337171949591172994570780693986, −20.54366771056582223845111203338, −19.65840224163009644100438348144, −18.814042187851378437897972801946, −17.20181212591982374111810205691, −16.38504810506003745138982305052, −15.686758488618046210880542945192, −14.65252543324152620991597485417, −14.16917677674629153632850935913, −13.11013308019631382054222742997, −11.82097720916488557681590880584, −11.36150735035705836766961782389, −10.09978915762941839069482095640, −9.54281296300669948411452129184, −8.07595033996045048760085205645, −6.87418155452804055025457452306, −5.6280641851661504463308620056, −4.942138716137916749734779952120, −3.82090974117708174111836359549, −3.10851950338303612033231580022, −1.70821648221569449127364499629, 1.22497840610763525889991486584, 2.640470087656160415475105889685, 3.350227498880993662545082493086, 4.97008606613088009728944367231, 5.749944779811133557727076208057, 6.83182439217917714071558713804, 7.60154789358645925266194940931, 8.447370905186766112957766103895, 10.03030338596438316456775636589, 11.296512865095134888696374293495, 12.063409600989762707092444480353, 12.887788382765342537690259851649, 13.574244776218770421445626218585, 14.52156711090953134129077639662, 15.22049828798135391848609286555, 16.46776104982197110906523164057, 17.24467199741684604186983972336, 18.28976403956042667291063938369, 19.234049469745575598122062014838, 20.29056429835519905117241040678, 20.72383990721403662275410231673, 22.18918163478257771064029732474, 22.672345431617510715187851724899, 23.6136155023600098808162856336, 24.32979122649481327596393128697

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