Properties

Label 1-385-385.202-r0-0-0
Degree $1$
Conductor $385$
Sign $-0.628 - 0.778i$
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  + (−0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s i·12-s + (−0.951 + 0.309i)13-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (0.587 + 0.809i)18-s + (−0.809 − 0.587i)19-s i·23-s + (0.309 + 0.951i)24-s + (0.809 − 0.587i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s i·12-s + (−0.951 + 0.309i)13-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (0.587 + 0.809i)18-s + (−0.809 − 0.587i)19-s i·23-s + (0.309 + 0.951i)24-s + (0.809 − 0.587i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.628 - 0.778i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ -0.628 - 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2777121720 - 0.5811960859i\)
\(L(\frac12)\) \(\approx\) \(0.2777121720 - 0.5811960859i\)
\(L(1)\) \(\approx\) \(0.6553686120 - 0.2488228696i\)
\(L(1)\) \(\approx\) \(0.6553686120 - 0.2488228696i\)

Euler product

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

−25.09980785797262828352674841343, −24.33293742937066337100562875990, −22.89912243417044864960093408676, −21.6245037152292643634923834098, −21.46568687078768617063191397788, −20.12071380542753394838236833430, −19.776108161533548198230671380820, −18.89770218027059223960603680008, −17.67345533536038224560582326860, −17.00283527211803551531124684673, −15.99989117734750236785181827584, −15.29391681879035933917860672608, −14.39024589543331984613383545797, −13.0995422465208151246582698540, −12.07441315347451888474363694748, −10.90679715445618855027883119810, −10.26698753151595777950132224553, −9.37693544987063520699434103410, −8.556186193277199833923489454240, −7.72533924944538980603961497813, −6.58244541941763447517247209606, −5.07758351537305404098007043089, −3.8426789188273058237729329544, −2.81630291876660904516742522861, −1.78619788010156418996308645045, 0.448994448014859962234151127613, 2.045709394923979556056969662538, 2.6685420297043904688124997645, 4.4934550546012452346852868891, 6.06584624709271066897012088794, 6.88220387090971772856910972244, 7.64290763375792818484310615217, 8.68400865509975913731655374579, 9.29081108058359972537211638250, 10.453114535570992539259263503352, 11.55489943102639533066644313043, 12.45477545690917414752350566031, 13.574054359747826598717398428890, 14.617469658897360497942853955629, 15.237994837754901568170196808029, 16.411011920163509482055502828423, 17.42378573619467900628526476301, 17.98997134791794904618916031342, 19.04604374544756130264599075912, 19.54487090052960712851827823164, 20.351606173840951495960867502238, 21.28655376003595390171992701011, 22.65332113119264319808382208967, 23.781203996200559888082650453839, 24.44824077672616283603383156852

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