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Properties

Label 1-384-384.269-r1-0-0
Degree $1$
Conductor $384$
Sign $0.146 - 0.989i$
Analytic cond. $41.2665$
Root an. cond. $41.2665$
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  + (0.980 − 0.195i)5-s + (−0.382 − 0.923i)7-s + (−0.555 + 0.831i)11-s + (0.980 + 0.195i)13-s + (−0.707 − 0.707i)17-s + (0.195 − 0.980i)19-s + (0.923 + 0.382i)23-s + (0.923 − 0.382i)25-s + (0.555 + 0.831i)29-s − i·31-s + (−0.555 − 0.831i)35-s + (−0.195 − 0.980i)37-s + (−0.923 − 0.382i)41-s + (−0.831 − 0.555i)43-s + (0.707 + 0.707i)47-s + ⋯
L(s)  = 1  + (0.980 − 0.195i)5-s + (−0.382 − 0.923i)7-s + (−0.555 + 0.831i)11-s + (0.980 + 0.195i)13-s + (−0.707 − 0.707i)17-s + (0.195 − 0.980i)19-s + (0.923 + 0.382i)23-s + (0.923 − 0.382i)25-s + (0.555 + 0.831i)29-s − i·31-s + (−0.555 − 0.831i)35-s + (−0.195 − 0.980i)37-s + (−0.923 − 0.382i)41-s + (−0.831 − 0.555i)43-s + (0.707 + 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.146 - 0.989i$
Analytic conductor: \(41.2665\)
Root analytic conductor: \(41.2665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (1:\ ),\ 0.146 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.555733396 - 1.341984856i\)
\(L(\frac12)\) \(\approx\) \(1.555733396 - 1.341984856i\)
\(L(1)\) \(\approx\) \(1.194337800 - 0.2870755416i\)
\(L(1)\) \(\approx\) \(1.194337800 - 0.2870755416i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.980 - 0.195i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (-0.555 + 0.831i)T \)
13 \( 1 + (0.980 + 0.195i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (0.195 - 0.980i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (0.555 + 0.831i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.195 - 0.980i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
43 \( 1 + (-0.831 - 0.555i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.555 - 0.831i)T \)
59 \( 1 + (-0.980 + 0.195i)T \)
61 \( 1 + (0.831 - 0.555i)T \)
67 \( 1 + (0.831 - 0.555i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.195 - 0.980i)T \)
89 \( 1 + (0.923 - 0.382i)T \)
97 \( 1 - iT \)
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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

−24.85805627779640183507081631930, −23.57763877334064111045471192240, −22.67639946775646900688002873207, −21.68633846533057214262681221622, −21.28321718965976592805568767985, −20.26848024236678854324237461381, −18.86146283282611136852168568761, −18.54860428668420659379538261758, −17.541254969083832781167054271561, −16.53212721166821409411258808646, −15.64214261760491281306737809320, −14.76287860776523322108688894239, −13.57315301322371276874694156990, −13.10123440204049838691905179435, −11.94495939080973346631809077110, −10.780444224016876736465492739695, −10.076358769460320535093109866309, −8.84884891947978730096512329126, −8.31903144897095553617932335682, −6.58362800914599046691291385907, −5.98978904708914877997646953960, −5.10448264391183010723330884216, −3.43760516781283782081072162646, −2.533426447044540681805691248784, −1.28617949791241060677477589817, 0.587050115055144969936025366986, 1.866852651304794430937870443, 3.07941434279198471556445528316, 4.44560509123928642472004848993, 5.349352197190586229555907833382, 6.642254180911670647110170653902, 7.25186874541914495541705400476, 8.75634201050320288548776395553, 9.54750960706945988571496158788, 10.463030608392245618759671536415, 11.27601718600615809464697545703, 12.76898416672934736225236797874, 13.39091367645610683382815694019, 14.00747129199609713831683121473, 15.33492663537324483827941108959, 16.17464256576237193447539217845, 17.1877028080013269823549938962, 17.821747932992794019721996544531, 18.696346159520533139583261443273, 20.03333432376309051747536803212, 20.5260551977862999839806659570, 21.41083592669187495018837143157, 22.41536529115604147582586991275, 23.23355544429883301222181305995, 24.02748939524298483238410971999

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