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Properties

Label 1-24e2-576.275-r0-0-0
Degree $1$
Conductor $576$
Sign $-0.825 - 0.564i$
Analytic cond. $2.67493$
Root an. cond. $2.67493$
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  + (−0.793 − 0.608i)5-s + (0.258 + 0.965i)7-s + (−0.991 − 0.130i)11-s + (0.130 + 0.991i)13-s − i·17-s + (−0.923 − 0.382i)19-s + (0.258 − 0.965i)23-s + (0.258 + 0.965i)25-s + (−0.608 − 0.793i)29-s + (−0.5 − 0.866i)31-s + (0.382 − 0.923i)35-s + (0.923 − 0.382i)37-s + (−0.258 + 0.965i)41-s + (−0.991 − 0.130i)43-s + (−0.866 − 0.5i)47-s + ⋯
L(s)  = 1  + (−0.793 − 0.608i)5-s + (0.258 + 0.965i)7-s + (−0.991 − 0.130i)11-s + (0.130 + 0.991i)13-s − i·17-s + (−0.923 − 0.382i)19-s + (0.258 − 0.965i)23-s + (0.258 + 0.965i)25-s + (−0.608 − 0.793i)29-s + (−0.5 − 0.866i)31-s + (0.382 − 0.923i)35-s + (0.923 − 0.382i)37-s + (−0.258 + 0.965i)41-s + (−0.991 − 0.130i)43-s + (−0.866 − 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.825 - 0.564i$
Analytic conductor: \(2.67493\)
Root analytic conductor: \(2.67493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 576,\ (0:\ ),\ -0.825 - 0.564i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09762750799 - 0.3156272678i\)
\(L(\frac12)\) \(\approx\) \(0.09762750799 - 0.3156272678i\)
\(L(1)\) \(\approx\) \(0.7013379246 - 0.07804628950i\)
\(L(1)\) \(\approx\) \(0.7013379246 - 0.07804628950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.793 - 0.608i)T \)
7 \( 1 + (0.258 + 0.965i)T \)
11 \( 1 + (-0.991 - 0.130i)T \)
13 \( 1 + (0.130 + 0.991i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.923 - 0.382i)T \)
23 \( 1 + (0.258 - 0.965i)T \)
29 \( 1 + (-0.608 - 0.793i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.923 - 0.382i)T \)
41 \( 1 + (-0.258 + 0.965i)T \)
43 \( 1 + (-0.991 - 0.130i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (0.793 + 0.608i)T \)
61 \( 1 + (-0.608 - 0.793i)T \)
67 \( 1 + (-0.991 + 0.130i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (-0.793 + 0.608i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
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

−23.531355132710488734348600882927, −22.97455516899229582917736374616, −21.93906426371979998177429350911, −21.01434949020362678416756353908, −20.12172628341355222372096501769, −19.51808339768985873813311207431, −18.54533389028333224686561206655, −17.76751119830157693910450578, −16.89349684836676459093646715666, −15.8943425967889868531012295029, −15.0897648260141911730241017901, −14.47224162740215941771444782612, −13.231850978828496092874609615605, −12.67341288539445982850571801930, −11.38984890699836545201625325562, −10.58786849007449540053377453843, −10.22870061397012258852881813546, −8.56656715874697042806949283182, −7.78422569162391937442583114989, −7.186028243558573157580461202028, −5.99549231719225820405335782709, −4.83294645781762098952230567529, −3.78234596475490251487685461730, −3.03168566259814041300864719077, −1.528747460586742036486478848225, 0.16549457707153732435172332777, 1.928923624621415125562437536729, 2.9161227333381010008024437877, 4.329516987692551360390576882004, 4.97120488657103013582025018437, 6.06013924455365261348199875542, 7.24749122594886565735266359405, 8.23812791583233470897093340370, 8.84873716412460138904920185249, 9.803177460483517059606138940455, 11.27798195576359126390637399093, 11.58048350261718568017607921606, 12.71745259326141877564819837387, 13.31624979710878649345342052315, 14.68986413216999699642406068844, 15.291715141531094723203241601331, 16.22949800879578831994993631470, 16.740321260351479879963786308821, 18.139363777430401055897695651503, 18.69560157962306587040292069671, 19.45310613498906955608335482281, 20.59013105073398767860553643729, 21.08279482628604130229870490736, 21.98940842006081084499752086742, 23.04502797957866732927634420807

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