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Properties

Label 1-384-384.5-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: $\chi_{384} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (1:\ ),\ -0.146 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2903861942 - 0.3366382996i\)
\(L(\frac12)\) \(\approx\) \(0.2903861942 - 0.3366382996i\)
\(L(1)\) \(\approx\) \(0.7233216610 + 0.06430149903i\)
\(L(1)\) \(\approx\) \(0.7233216610 + 0.06430149903i\)

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

−24.36847864785586571377102552293, −23.71982781532015456647107254338, −22.61682291697189446564506735051, −22.3357571894785264975729090054, −20.86551898546172713185352602334, −20.07994960156283210556698706543, −19.27692231071696426662389540074, −18.71330052584409320115789408257, −17.21288966620538836336472503163, −16.69473903004965606115854416875, −15.69203750944781097922999241496, −14.79829291655625694081826148007, −13.86417172813551070858201696704, −12.92083914805313330484237276102, −11.79723703733474418589557769674, −11.11941372388911928674719165554, −10.08628098245124567771458940781, −9.03635612781688158798397370374, −7.81476906177401837529950168780, −7.178201891951699121216899909650, −6.10280102165817576184150382010, −4.61777605294124079939873432785, −3.784183782732430127681079114, −2.78318572131226737466804562743, −0.92905573086665653756760292579, 0.15331020078781767469814445584, 1.926711523000824407676331889386, 3.105948027741878905489438998445, 4.35364241838050745673312782931, 5.16409224101688911611905007430, 6.67152356753093813902048244333, 7.31972392260248296011608085052, 8.73486396295893546763275979889, 9.17725696998074529892881428707, 10.553445145609602593044589240, 11.60480560846156485892054749612, 12.39373744045759142885515623281, 12.97925169546239391357915037502, 14.657244408834454764223998156365, 15.11441068648745208809552185570, 15.97349073672107200105779894759, 16.99365684465151894792617688632, 17.88209817641062213435203541229, 19.06805560207022263971063534900, 19.59480574798818925054242755352, 20.34998078791059875979106164782, 21.695045401799268547481556093115, 22.25789336038987322842096906764, 23.18531558292585613354769706979, 24.11308364889662919693855523331

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