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Properties

Label 1-385-385.19-r0-0-0
Degree $1$
Conductor $385$
Sign $0.986 - 0.161i$
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  + (−0.978 + 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 − 0.866i)12-s + (−0.309 + 0.951i)13-s + (0.669 − 0.743i)16-s + (0.978 + 0.207i)17-s + (0.913 − 0.406i)18-s + (0.913 + 0.406i)19-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)24-s + (0.104 − 0.994i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 − 0.866i)12-s + (−0.309 + 0.951i)13-s + (0.669 − 0.743i)16-s + (0.978 + 0.207i)17-s + (0.913 − 0.406i)18-s + (0.913 + 0.406i)19-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)24-s + (0.104 − 0.994i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.986 - 0.161i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.986 - 0.161i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7769165431 - 0.06304344582i\)
\(L(\frac12)\) \(\approx\) \(0.7769165431 - 0.06304344582i\)
\(L(1)\) \(\approx\) \(0.6975559265 - 0.08719354767i\)
\(L(1)\) \(\approx\) \(0.6975559265 - 0.08719354767i\)

Euler product

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

−24.96190733085235594216527121562, −23.67021218384082102884020943105, −22.52728439357354982285256276571, −21.80682890527425665222016850708, −20.74805294017834367682237031372, −20.324438121053192759884206397, −19.34642327649795952157891242446, −18.302009339080214511226347319089, −17.40458072643902954265695472513, −16.69271692409276234478395722723, −15.80977079707145964888419223273, −15.14272305108236920850624935695, −14.056942506910997102817310131847, −12.5079563385394401538144753898, −11.708673448180407400662073342147, −10.644708973033166660336473748980, −10.0834430776540331129009246262, −9.18064907955870199234976316519, −8.274626878292379062161846451297, −7.26398909960580150800915307632, −5.94414479530037911881782615758, −4.91290496297424698030413757163, −3.42187827871113817174856796519, −2.67972201329970524553840234390, −0.82757268802535654051534274281, 1.06272084045599594696856375121, 2.029316690471488084354667700008, 3.280290615517302736447340182653, 5.28587955733763247057168855764, 6.192942133426050768307698806086, 7.24135171239830418178466155805, 7.77706233087288434922753841317, 8.921281805999402249625899793704, 9.76128447635704162087494363725, 11.02027203237220266535076668610, 11.80267242827350965485523659735, 12.598225051027779036124134339993, 13.97860575829752096634679318870, 14.6188795666421070249480612763, 15.9551446339018542305456721252, 16.785199110576650809302269448910, 17.507200766977399414158252000802, 18.44761283036670349480557154794, 19.02390901510290992069619069796, 19.7878775500254522401300591494, 20.70793952703769674951730407564, 21.79348014486501862220774469482, 23.1726411459402714504983035953, 23.79198736831550632497429366232, 24.60897607201208628944522224421

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