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Properties

Label 1-385-385.258-r0-0-0
Degree $1$
Conductor $385$
Sign $0.281 + 0.959i$
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  + (0.587 + 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.809 + 0.587i)6-s + (−0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + i·12-s + (0.587 + 0.809i)13-s + (−0.809 − 0.587i)16-s + (0.587 − 0.809i)17-s + (0.951 + 0.309i)18-s + (0.309 + 0.951i)19-s + i·23-s + (−0.809 + 0.587i)24-s + (−0.309 + 0.951i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.809 + 0.587i)6-s + (−0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + i·12-s + (0.587 + 0.809i)13-s + (−0.809 − 0.587i)16-s + (0.587 − 0.809i)17-s + (0.951 + 0.309i)18-s + (0.309 + 0.951i)19-s + i·23-s + (−0.809 + 0.587i)24-s + (−0.309 + 0.951i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.281 + 0.959i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (258, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.281 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.908413680 + 1.429252913i\)
\(L(\frac12)\) \(\approx\) \(1.908413680 + 1.429252913i\)
\(L(1)\) \(\approx\) \(1.664894280 + 0.7889431126i\)
\(L(1)\) \(\approx\) \(1.664894280 + 0.7889431126i\)

Euler product

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

−24.329943813981550171258716570770, −23.31753422763677395620389215228, −22.38723583408815179993652959526, −21.572644572606694941489999186019, −20.76524636381430543653073981124, −20.17931749324943612192450613328, −19.280651901821514386363766771, −18.615376035884172581443704707819, −17.477932042118061728126104741828, −15.96550673574667185841013067476, −15.214222201014299700644129194, −14.461186142483513867799075805352, −13.49404475875630173659578213050, −12.91864018065624089449344889831, −11.80291755507400330205565605718, −10.58353960778708389360658054768, −10.057838342627127327124997220079, −8.90591323671823774930463423681, −8.10747686260297608753566454579, −6.63327437830576764382109181525, −5.3721812209866074946373223560, −4.33136776491714212172039069170, −3.365923033738975875367384616880, −2.54768246829990529199897353053, −1.25727861244520368661238133395, 1.67584107415128496132062634117, 3.13714474091856706197421800814, 3.84355189549001748946790961365, 5.09007293545753241909393526875, 6.29542815349573602265793687397, 7.23925419690296817798520739245, 8.0071766721917890556715856373, 8.96315330736702708448747397509, 9.797351459433105719583774332772, 11.53322088171535801059900474255, 12.40087221483076775882463053912, 13.411322990871585295452030878585, 14.0621224158172385345940722361, 14.72033335446333623813765279058, 15.80963625271738163399079147642, 16.39872369633286559198559318693, 17.669921886896838575631762924834, 18.51966168961889780817130590593, 19.338698780221370449955038295730, 20.78152044137256109031911697154, 20.96445500843397289914219130700, 22.18419914178867088436611599856, 23.17270411582998917166624588316, 23.92867536439033781519002383073, 24.71655953029448304967697208505

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