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Properties

Label 1-385-385.128-r0-0-0
Degree $1$
Conductor $385$
Sign $-0.898 + 0.439i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.898 + 0.439i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ -0.898 + 0.439i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4622423293 + 1.996600675i\)
\(L(\frac12)\) \(\approx\) \(0.4622423293 + 1.996600675i\)
\(L(1)\) \(\approx\) \(1.100769724 + 1.229455452i\)
\(L(1)\) \(\approx\) \(1.100769724 + 1.229455452i\)

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

−23.96876791733195820143822452383, −23.33727855763108068455974532722, −22.480741625106057884856818269684, −21.45030141536963182181803542707, −20.51775855614582400758334011077, −19.92359925706174835044802858669, −18.95565686057625409880997402875, −18.36350676807684279529800780602, −17.34753465950373923804427311369, −15.84864216294106984576584730323, −15.01732878273053962320317792729, −14.00015091349055391516737474223, −13.321857161976886274973559899460, −12.73237741316207534316134732738, −11.45613752159187797386068250903, −11.096568788198548028233633891506, −9.464809188866400924641353615533, −8.79060520891107427520418859628, −7.30729459844983959528902022653, −6.49042577267019946611652360216, −5.46061830933589246397848401210, −4.16604215193952653662156594448, −3.05624279050084998123051561167, −2.116262137321128825460849885565, −0.93706260875797035673423288591, 2.223876098488138618701306724538, 3.52955464870168001155334140424, 4.10256158222268406103172419877, 5.31343221707838568733205933716, 6.101440270396741085364214165714, 7.42042936318729194326531010643, 8.46894679105274423506641197819, 9.10512498549700073255045363180, 10.58082150654107536914421886460, 11.2722149631820871569377490805, 12.680599759392409225157600132189, 13.40084189819276241971040151082, 14.51363183341567481495449345788, 14.99230052409649280810682759125, 15.99159123244946314328414719197, 16.566375410797836928699953271080, 17.53852566867913828126171073392, 18.701183389329725492526854258253, 20.05699061822672966023098968249, 20.73604576343272205981832066037, 21.475329880332559438435222803792, 22.33828023863299977292687548080, 23.00528062225159742185042935973, 23.9671055688860242975255046076, 25.04549355990429119719128797379

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