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Properties

Label 1-385-385.338-r0-0-0
Degree $1$
Conductor $385$
Sign $0.00178 - 0.999i$
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  + (−0.207 − 0.978i)2-s + (0.994 − 0.104i)3-s + (−0.913 + 0.406i)4-s + (−0.309 − 0.951i)6-s + (0.587 + 0.809i)8-s + (0.978 − 0.207i)9-s + (−0.866 + 0.5i)12-s + (−0.951 − 0.309i)13-s + (0.669 − 0.743i)16-s + (0.207 − 0.978i)17-s + (−0.406 − 0.913i)18-s + (0.913 + 0.406i)19-s + (0.866 − 0.5i)23-s + (0.669 + 0.743i)24-s + (−0.104 + 0.994i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (0.994 − 0.104i)3-s + (−0.913 + 0.406i)4-s + (−0.309 − 0.951i)6-s + (0.587 + 0.809i)8-s + (0.978 − 0.207i)9-s + (−0.866 + 0.5i)12-s + (−0.951 − 0.309i)13-s + (0.669 − 0.743i)16-s + (0.207 − 0.978i)17-s + (−0.406 − 0.913i)18-s + (0.913 + 0.406i)19-s + (0.866 − 0.5i)23-s + (0.669 + 0.743i)24-s + (−0.104 + 0.994i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00178 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00178 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.00178 - 0.999i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (338, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.00178 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078271809 - 1.080199966i\)
\(L(\frac12)\) \(\approx\) \(1.078271809 - 1.080199966i\)
\(L(1)\) \(\approx\) \(1.071844887 - 0.6288686563i\)
\(L(1)\) \(\approx\) \(1.071844887 - 0.6288686563i\)

Euler product

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

−24.63072736953389793826017850358, −24.2818688673582031799621327167, −23.2076612144225352860399995452, −22.07755857540790969236221554378, −21.41556760376668871881202125837, −20.13671158197161444140740831220, −19.35039147188835468987944818642, −18.66519792824372671765573611049, −17.58285294957059302952137131789, −16.703842165465735701998090066876, −15.7501966337080647799020400793, −14.90332461450370595218331733310, −14.368058417987421622023421394277, −13.35092855823495724902513423908, −12.60299780983394245854181090052, −10.962461240937713002336809005785, −9.60340685030739298408880104272, −9.34715164494938334610679719236, −8.021468928280629694300970407755, −7.50298201944516217869423642731, −6.413201801377433677126460285545, −5.10189171524465064852574177744, −4.1735947739981725537242364646, −2.949602418755002106674120194311, −1.41281033370804867438697882363, 1.03511593678960955321794470552, 2.41087747819895442289278532700, 3.07583333195756224911854687180, 4.24043593139308135018986135927, 5.28296209630078842610487413065, 7.19483216927904692569016969504, 7.89673175206435391310259914959, 9.040901280151550203979313929307, 9.66492799457076519899205226205, 10.552974107061015999169942920671, 11.83269854665458108014038641207, 12.56625940749226308241289568377, 13.55562881353836632754104035443, 14.23896328090833808259270615829, 15.18738228555385375325662966608, 16.43873466904430645132775738442, 17.53178035980678217949851790777, 18.52712356329016330645241784508, 19.07874111955505693333844099523, 20.10718211121015227786631549169, 20.536644839991081737335678140423, 21.44963112655105606827709785720, 22.344189091587832664302302375673, 23.2055472822306077937059833805, 24.54551202926549519188567747022

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