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Properties

Label 1-385-385.284-r0-0-0
Degree $1$
Conductor $385$
Sign $0.981 + 0.190i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.981 + 0.190i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.981 + 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.284081501 + 0.1235378253i\)
\(L(\frac12)\) \(\approx\) \(1.284081501 + 0.1235378253i\)
\(L(1)\) \(\approx\) \(1.047350569 + 0.08787297863i\)
\(L(1)\) \(\approx\) \(1.047350569 + 0.08787297863i\)

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

−24.87098021125504563604910573349, −23.910703603575424398571970813003, −22.44482794573764741047904553453, −21.513699716478085546693093521288, −20.809184868035017786235531926316, −19.937482685219633648722811700857, −19.46074402948813557120612642458, −18.37549378401845133755052659553, −17.724117723417677055807448667301, −16.54301906948711795036736224840, −15.57427976458151705119422042159, −15.06804178191759129301206825386, −13.452069659017418930549046602720, −13.079005495864695931264693131418, −11.59944320313539526743058809200, −10.796214880254137984275737353529, −9.7444867837576316034118684923, −9.05746064263946189856696221061, −8.14487422009846889976872479032, −7.408061869969849796323595682220, −6.18480713185561562295782716958, −4.41558648633032203608048562738, −3.33278350394472142502903124286, −2.45260228445442003798888260349, −1.209280113977964819622294962556, 1.222186599247457699445945878631, 2.27820294794703419008840497921, 3.50292232322546384623544523670, 4.968662708630507319320246892973, 6.48272907758189472005190148020, 7.09016871118979906327708607107, 8.28865731622871619933574071084, 8.8265360817850250558960711993, 9.73809997454992875227525289208, 10.729936769536436906000834178305, 11.81126069370204761969414259252, 13.08601307830813738541074467492, 14.12657543004494456865762713492, 14.75962861098082235206600924334, 15.88926518747473336751867285709, 16.37119166793671649424823626419, 17.781648407524232766341132899952, 18.43769577857757365187746525132, 19.12896967497136525125980709166, 20.15512370214749580948921070655, 20.59908631679363540551911568534, 21.688087329243493799738740633066, 23.12809401358288560340850972600, 23.97434502758760267621191297268, 24.85640273612117915488023904840

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