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Properties

Label 1-3024-3024.2477-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.187 + 0.982i$
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  + (0.342 + 0.939i)5-s + (0.342 − 0.939i)11-s + (0.642 − 0.766i)13-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.173 − 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.642 − 0.766i)29-s + (−0.173 + 0.984i)31-s + (0.866 + 0.5i)37-s + (−0.766 − 0.642i)41-s + (−0.342 + 0.939i)43-s + (0.173 + 0.984i)47-s − i·53-s + 55-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)5-s + (0.342 − 0.939i)11-s + (0.642 − 0.766i)13-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.173 − 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.642 − 0.766i)29-s + (−0.173 + 0.984i)31-s + (0.866 + 0.5i)37-s + (−0.766 − 0.642i)41-s + (−0.342 + 0.939i)43-s + (0.173 + 0.984i)47-s − i·53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.186958664 + 0.9813561747i\)
\(L(\frac12)\) \(\approx\) \(1.186958664 + 0.9813561747i\)
\(L(1)\) \(\approx\) \(1.076474823 + 0.2228198774i\)
\(L(1)\) \(\approx\) \(1.076474823 + 0.2228198774i\)

Euler product

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

−18.87937157683352524419660808895, −18.08546627759460195054985917884, −17.48551886055723744362723387829, −16.77228348572914689048222718690, −16.25249097095769471380960375970, −15.3955286157072123423840271629, −14.78655836273106611058747182498, −13.810395032159352799396388276642, −13.22634768094802588346611915426, −12.72254198324079155018962787304, −11.690371916312973629165571199487, −11.36627642100008570140935779013, −10.22766636598632945211443813878, −9.39404661569501713989061153876, −9.07203822398618289250660789715, −8.26057934760735171108817856640, −7.24545403867501975374385566334, −6.65260669128992585006805144003, −5.72130338303155319946411031335, −4.910959786460010786052789865095, −4.3199600044406009032731046635, −3.50566722004636657770376516491, −2.09602549471661961839189232500, −1.76402789404747649228404547772, −0.50374181602101197992089946778, 1.02446187694474170708521375802, 2.05082678183321324622544389911, 2.91117566920621823805005309573, 3.62125023671742393465748955709, 4.38978222576450480069515922266, 5.66118379460239177708497906326, 6.17743268189654208735764242359, 6.685285278684984896277628012867, 7.78886418516525143532063021617, 8.4339732741139916226219268048, 9.12222708816062406177398061755, 10.2385170431334903808011366739, 10.6525632013684032253817815566, 11.207470887441469437896632856997, 12.11299229989478433170388700637, 13.11557987916647911570249055491, 13.45795122604153899145982899871, 14.51462107492882655867621562548, 14.813606015551566457519269329865, 15.645787232509917850170820823776, 16.46287246109522988890472276747, 17.192429443913398556692558463241, 17.822798660040981418449323512696, 18.60943553862748090484526919000, 19.05378317017705905627453785436

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