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Properties

Label 1-387-387.85-r1-0-0
Degree $1$
Conductor $387$
Sign $-0.939 + 0.342i$
Analytic cond. $41.5889$
Root an. cond. $41.5889$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 8-s + 10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − 19-s + (0.5 − 0.866i)20-s + (0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 8-s + 10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − 19-s + (0.5 − 0.866i)20-s + (0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(41.5889\)
Root analytic conductor: \(41.5889\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 387,\ (1:\ ),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1978834509 - 1.122252817i\)
\(L(\frac12)\) \(\approx\) \(-0.1978834509 - 1.122252817i\)
\(L(1)\) \(\approx\) \(0.9558535299 - 0.6422557311i\)
\(L(1)\) \(\approx\) \(0.9558535299 - 0.6422557311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−24.71846116878039726835909832485, −23.88575831778837271148795198844, −23.452010073145567103846325148044, −21.82111673849508542588938143736, −21.5394808461830123199958076882, −20.893387751706109024593813041761, −19.410303921935808775560109451464, −18.371054366606763344538797086176, −17.559755944333996158692939190, −16.578581459407913448670584434815, −16.099189148360397705945529905039, −14.93158969890159776268806042234, −14.15358728847560147861874354072, −13.298931063304550076285996199328, −12.33515720672826022020471911635, −11.69481779232681523965543685106, −10.06563977309246837129162648592, −8.76756144535923416079887259686, −8.51501460535392098320976223195, −7.227943879261837382485331354845, −5.91602813403112993430792470865, −5.36073618621234421637302558435, −4.45602660498272040538522168186, −3.05625463105303769451027093461, −1.667301468236092813217368200190, 0.24323419197989845377926174057, 1.79042663742870110510577015747, 2.67275098144325769865596978337, 3.84775775368110467265964005594, 4.87896292025067945520106191074, 5.913002023388473468234042756273, 7.09336539831058217602971346323, 8.153037186459086459284175784868, 9.82825233254053363532774922998, 10.275964513133322019374554064142, 10.95603099364344353341431807852, 12.17657426164024322775897446941, 13.007740724605023872963580242065, 13.98095096265509916127002972006, 14.66190907751517646631802235329, 15.36369482059438580925103104, 17.09352160140289828241308316013, 17.770396146990473225979179892248, 18.63061626031165044157534481231, 19.51381533138744459063018587664, 20.58879807728097234599082798827, 20.98920785781533834265323843454, 22.10115018023311982245515951114, 22.85934902546580101491054394861, 23.41427022237198215426777833153

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