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Properties

Label 1-385-385.142-r0-0-0
Degree $1$
Conductor $385$
Sign $-0.0932 + 0.995i$
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  + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s − 6-s + i·8-s + (0.5 + 0.866i)9-s + (0.866 − 0.5i)12-s + i·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)18-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)24-s + (−0.5 − 0.866i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s − 6-s + i·8-s + (0.5 + 0.866i)9-s + (0.866 − 0.5i)12-s + i·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)18-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)24-s + (−0.5 − 0.866i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0932 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0932 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.0932 + 0.995i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ -0.0932 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7737107342 + 0.8495312596i\)
\(L(\frac12)\) \(\approx\) \(0.7737107342 + 0.8495312596i\)
\(L(1)\) \(\approx\) \(0.8624655288 + 0.4471924152i\)
\(L(1)\) \(\approx\) \(0.8624655288 + 0.4471924152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + iT \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 - T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + iT \)
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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

−24.64105833145871090825696265299, −23.56046491700381696952429967328, −22.374719639823191612905096128355, −21.29206673565008106546243472645, −20.4443182736394052580974194264, −20.01115812792988926333523542239, −18.87586076962606295684106656544, −18.445040547745417370414361092718, −17.49868824349110746204448503834, −16.472064897413647220457035677022, −15.46004779201657798761752647000, −14.486016496686996957276142009775, −13.37992447165177978727621876919, −12.49165191810880433231381268363, −11.81001971009445371853214104421, −10.35211251225032595073101134731, −9.81106870214376878431185186465, −8.582169863957092078529858646419, −8.00685905952676955004344291079, −7.142508952298919249641702437017, −5.93809749852420089688074562547, −4.0507120261266061564312095141, −3.06317421516655570758791645201, −2.13317136083725341833756906986, −0.8752484868751711554227626938, 1.52942682820151438858880222217, 2.61801359495751077843393731441, 4.03271143546374597822973967740, 5.18983693091600814479398359852, 6.48545830737205112933298816417, 7.48902773392726884859417353383, 8.406387534406179484725269524188, 9.15716574814700171781250399842, 10.001613789529682293279686625048, 10.82323365184995043726367865267, 11.99449153682271685523314514948, 13.50090605493684398318507040379, 14.35069275652124637227650819223, 15.044845849881353177862908802636, 16.01962457330505330166859995497, 16.61161516868465430789344240429, 17.703849534973629738803636477697, 18.721983851723395005533432815568, 19.49735882735277245859943261175, 20.05621575747597841540106878474, 21.2247817549498210589407018522, 21.79531053045546379997845326950, 23.39350867882734157971950017653, 23.98741006268091568411136284796, 25.10553355345448255527357780805

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