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Properties

Label 1-385-385.228-r0-0-0
Degree $1$
Conductor $385$
Sign $0.794 - 0.606i$
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  + (−0.743 + 0.669i)2-s + (−0.406 + 0.913i)3-s + (0.104 − 0.994i)4-s + (−0.309 − 0.951i)6-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.866 + 0.5i)12-s + (−0.951 − 0.309i)13-s + (−0.978 − 0.207i)16-s + (0.743 + 0.669i)17-s + (0.994 + 0.104i)18-s + (−0.104 − 0.994i)19-s + (−0.866 − 0.5i)23-s + (−0.978 + 0.207i)24-s + (0.913 − 0.406i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (−0.406 + 0.913i)3-s + (0.104 − 0.994i)4-s + (−0.309 − 0.951i)6-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.866 + 0.5i)12-s + (−0.951 − 0.309i)13-s + (−0.978 − 0.207i)16-s + (0.743 + 0.669i)17-s + (0.994 + 0.104i)18-s + (−0.104 − 0.994i)19-s + (−0.866 − 0.5i)23-s + (−0.978 + 0.207i)24-s + (0.913 − 0.406i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.794 - 0.606i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (228, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.794 - 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3659646578 - 0.1237463056i\)
\(L(\frac12)\) \(\approx\) \(0.3659646578 - 0.1237463056i\)
\(L(1)\) \(\approx\) \(0.5050543209 + 0.1714817661i\)
\(L(1)\) \(\approx\) \(0.5050543209 + 0.1714817661i\)

Euler product

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

−24.72522700598811102130948880831, −23.8013575153742659771243448720, −22.70676793046108892164889718141, −22.01789415473742836787753545944, −20.95413120843455004809416565296, −19.97653945803311168396727806938, −19.269731996431257042286081982280, −18.43594315064176254919813708945, −17.859283177931881845838229504277, −16.74320200385881254559421210673, −16.37617490384016028316913547996, −14.654241039171863300978194169518, −13.64232914370123143424285293569, −12.59852481238512107625792910257, −12.01014859684292641781165283804, −11.20905292495088040082827182199, −10.112742669262735931374268000076, −9.22916253957540544717082786904, −7.89268399007786291028565975562, −7.464759776007460716860596607, −6.268925677972315389358039811980, −4.988236638376292419058909094864, −3.474165884279147824368981791234, −2.2693613922250804043467952195, −1.31902452404346789771899678044, 0.32130155857323224741005313612, 2.211488056817129083432160013429, 3.83415045163642527647138303188, 5.056638172363381484439302417666, 5.7570755344652115392648310524, 6.88908856295422892470061259253, 7.97403669440866357584677601406, 9.030808893095095193188802536818, 9.83766286162489195368658624374, 10.575743256980244656438188086213, 11.48087519549174185867491354307, 12.67587499130571871930787033889, 14.25536819150485258359674994931, 14.85051337011156111532681653583, 15.72180614992980158536464992072, 16.50971968525795835221052616405, 17.29874218655448140946018763021, 17.91388221340631749910110662761, 19.15252682823038299433228198343, 19.94191820179216759550306295883, 20.86046577130760286518850363910, 21.969320706223622062031401403583, 22.67871273847513303954652693541, 23.72009346009519660556220048134, 24.36643024209428940897366195095

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