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Properties

Label 1-384-384.155-r0-0-0
Degree $1$
Conductor $384$
Sign $0.989 - 0.146i$
Analytic cond. $1.78328$
Root an. cond. $1.78328$
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  + (0.195 − 0.980i)5-s + (−0.382 + 0.923i)7-s + (0.831 − 0.555i)11-s + (0.195 + 0.980i)13-s + (−0.707 + 0.707i)17-s + (0.980 − 0.195i)19-s + (0.923 − 0.382i)23-s + (−0.923 − 0.382i)25-s + (0.831 + 0.555i)29-s − i·31-s + (0.831 + 0.555i)35-s + (0.980 + 0.195i)37-s + (0.923 − 0.382i)41-s + (−0.555 − 0.831i)43-s + (−0.707 + 0.707i)47-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)5-s + (−0.382 + 0.923i)7-s + (0.831 − 0.555i)11-s + (0.195 + 0.980i)13-s + (−0.707 + 0.707i)17-s + (0.980 − 0.195i)19-s + (0.923 − 0.382i)23-s + (−0.923 − 0.382i)25-s + (0.831 + 0.555i)29-s − i·31-s + (0.831 + 0.555i)35-s + (0.980 + 0.195i)37-s + (0.923 − 0.382i)41-s + (−0.555 − 0.831i)43-s + (−0.707 + 0.707i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.989 - 0.146i$
Analytic conductor: \(1.78328\)
Root analytic conductor: \(1.78328\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (0:\ ),\ 0.989 - 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.373067694 - 0.1012835579i\)
\(L(\frac12)\) \(\approx\) \(1.373067694 - 0.1012835579i\)
\(L(1)\) \(\approx\) \(1.137818143 - 0.05991848213i\)
\(L(1)\) \(\approx\) \(1.137818143 - 0.05991848213i\)

Euler product

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

−24.88436810585693645457755915130, −23.2569391492877284115149951994, −22.875419453494148562649466650704, −22.162530017441445544266829702311, −21.050057760543942405039711162379, −19.929691644945282340141705703366, −19.5589159552040981452737835250, −18.118481078409650168826740359, −17.73915747345491373823179428084, −16.63576692948564565464377058418, −15.607177047033658653507764267547, −14.71756565970661498620836488639, −13.83776616762712849218274831334, −13.10991178536601149559368731584, −11.81130141443916253247495032205, −10.91387387935489415755083396735, −10.05968197832216326654337604804, −9.28828109782356302239182169436, −7.74107310355707348023760505334, −6.99953942266537163500903509583, −6.2003648892996284692698415681, −4.8048597693073956252580932573, −3.58207052879051283671268256279, −2.74746588197727021343929561460, −1.11925169234553115365874977501, 1.1333852721953821429227748183, 2.37763610552161543577585115157, 3.786296553199042309098695952395, 4.85834349178004628502626639877, 5.92354327662774953685433122155, 6.73454478668644238518677633115, 8.31892504099315228363222153367, 9.02353305499266508722914636320, 9.60769764097571304457020836288, 11.18941476426804203293997491747, 11.93709941647511112406117716860, 12.84017137590938189040789288545, 13.67658858311047333395131689915, 14.73384553106353781515486177746, 15.829870461189859813630047721743, 16.50219291717802495125429227680, 17.32470489197903206610323871886, 18.42637099090648234531087341391, 19.320748066467392654815535580709, 20.04365031296908860036260756786, 21.182586122999913607819590122341, 21.774341399681644534343497529998, 22.61290079934122086143391881882, 23.92066223522327383748003669037, 24.47357771963549926571374334964

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