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Properties

Label 1-3024-3024.2453-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.310 + 0.950i$
Analytic cond. $14.0433$
Root an. cond. $14.0433$
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  + (0.642 + 0.766i)5-s + (0.642 − 0.766i)11-s + (0.642 + 0.766i)13-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.939 − 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.642 + 0.766i)29-s + (−0.766 + 0.642i)31-s + i·37-s + (−0.766 + 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.766 + 0.642i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)5-s + (0.642 − 0.766i)11-s + (0.642 + 0.766i)13-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.939 − 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.642 + 0.766i)29-s + (−0.766 + 0.642i)31-s + i·37-s + (−0.766 + 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.766 + 0.642i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.310 + 0.950i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.310 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9489429972 + 1.308401971i\)
\(L(\frac12)\) \(\approx\) \(0.9489429972 + 1.308401971i\)
\(L(1)\) \(\approx\) \(1.117653064 + 0.3259310534i\)
\(L(1)\) \(\approx\) \(1.117653064 + 0.3259310534i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (0.642 - 0.766i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.642 + 0.766i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.984 + 0.173i)T \)
61 \( 1 + (-0.642 + 0.766i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.642 - 0.766i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−18.604130072933986693208238668236, −18.10304701828074193565513008916, −17.45471633333607094409834022850, −16.83161455725843534802769850431, −16.01465951011607863233105885483, −15.51450970407520873131230475421, −14.542410762658797776903397240, −13.822575592662672147484327289993, −13.27802175198113320856784511257, −12.49950210030663414466908138789, −11.8831146495414638567485362618, −11.09658254948664793018475970548, −10.10510360343456133719888077270, −9.50297551758535877710941494919, −9.02772359047741897387493414298, −8.00545974692665280726927080869, −7.42138713776790588278087992256, −6.3526540390883609327623445198, −5.71298175656357053069480112574, −5.01883516022366313883863716911, −4.14669842386191700412249583722, −3.376809832180466285984058696645, −2.14795206283077160749349623184, −1.58747943227699683092540287945, −0.46743139380725096795985196319, 1.329631932969986754644098586013, 1.90243709809342846930228714994, 3.066135068142369432626601736952, 3.61255502559987102940308260496, 4.5609455574236792392343139219, 5.647090723735379512499413375963, 6.25645552859209121511181128852, 6.80066126875960909352997716372, 7.66759226146257981840631520615, 8.77738280101192099990888426964, 9.11247854181718326406183141370, 10.10313927875504489987899989411, 10.79074708667807078156339644763, 11.360745044176797466206724604414, 12.09312713423994416186858510335, 13.12636248590828805792508275866, 13.80473813309799609132100673020, 14.18615410653021327677559081642, 14.99574808965785381868374240474, 15.75804065862060838044178095627, 16.59104056341083743267849533882, 17.11124602154776504216773812138, 18.04570513740954587228740304373, 18.462523277898215465288403456987, 19.14731149601377363222824121657

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