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Properties

Label 1-385-385.123-r0-0-0
Degree $1$
Conductor $385$
Sign $-0.735 + 0.676i$
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  + (0.406 + 0.913i)2-s + (−0.207 − 0.978i)3-s + (−0.669 + 0.743i)4-s + (0.809 − 0.587i)6-s + (−0.951 − 0.309i)8-s + (−0.913 + 0.406i)9-s + (0.866 + 0.5i)12-s + (−0.587 + 0.809i)13-s + (−0.104 − 0.994i)16-s + (−0.406 + 0.913i)17-s + (−0.743 − 0.669i)18-s + (0.669 + 0.743i)19-s + (−0.866 − 0.5i)23-s + (−0.104 + 0.994i)24-s + (−0.978 − 0.207i)26-s + (0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (−0.207 − 0.978i)3-s + (−0.669 + 0.743i)4-s + (0.809 − 0.587i)6-s + (−0.951 − 0.309i)8-s + (−0.913 + 0.406i)9-s + (0.866 + 0.5i)12-s + (−0.587 + 0.809i)13-s + (−0.104 − 0.994i)16-s + (−0.406 + 0.913i)17-s + (−0.743 − 0.669i)18-s + (0.669 + 0.743i)19-s + (−0.866 − 0.5i)23-s + (−0.104 + 0.994i)24-s + (−0.978 − 0.207i)26-s + (0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.735 + 0.676i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ -0.735 + 0.676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2998446890 + 0.7688974209i\)
\(L(\frac12)\) \(\approx\) \(0.2998446890 + 0.7688974209i\)
\(L(1)\) \(\approx\) \(0.8147595636 + 0.3985592439i\)
\(L(1)\) \(\approx\) \(0.8147595636 + 0.3985592439i\)

Euler product

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

−23.937381009506365679554524517618, −22.86984148937760132602794414882, −22.30807471762155187277738243929, −21.674446349773858156593622831138, −20.60471187567148968882685516260, −20.1625449206786054866347785768, −19.222664671189398385771452741767, −17.92703159547284768079882444903, −17.37669372046825752015005742761, −15.92337753264360621976215306480, −15.317817399184432267320597107559, −14.27901186946909922423690433235, −13.48931192825346506501208958359, −12.27245701849360157196160884464, −11.48354275447024302583292862658, −10.69578731611084482055406518383, −9.70098276784869073132640328989, −9.20904973161997897029686809861, −7.7844723739663602750751625485, −6.09326040558430571201946791416, −5.183210655163007778757843871981, −4.38805937785470746068069229464, −3.28846543255913086963608731521, −2.371696489276204283283410163999, −0.433866801662804624161392170902, 1.64520820586334937194605321337, 3.08181404848965528922993511944, 4.41804917167980176832872684747, 5.52086283923919451180642596073, 6.4479881328335512489628060847, 7.16959646899839869356058634241, 8.146600798532280500629838350818, 8.9386884812446038428491914015, 10.36009441062695594154407117316, 11.88736398064141379897986213067, 12.3263031299205378612029191108, 13.40000681925550136167025678268, 14.14414599468230105420335902205, 14.8834393705357315801708407396, 16.19525183950950052647573784187, 16.84150470952065032827935341421, 17.755880970208138768205026751414, 18.46101343355166209603816909689, 19.39134768306440834190613954308, 20.46878599779547563992569157570, 21.836060535024516045647887246579, 22.3028707689422340811121728742, 23.50686552011384919396875582685, 23.8836027693728176413091570145, 24.7228160910906178185283577830

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