Properties

Label 1-3024-3024.2651-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.998 - 0.0622i$
Analytic cond. $324.973$
Root an. cond. $324.973$
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 + (−0.984 + 0.173i)13-s + 17-s i·19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.984 + 0.173i)29-s + (0.766 − 0.642i)31-s + (−0.866 − 0.5i)37-s + (−0.173 − 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.766 − 0.642i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)5-s + (−0.342 − 0.939i)11-s + (−0.984 + 0.173i)13-s + 17-s i·19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.984 + 0.173i)29-s + (0.766 − 0.642i)31-s + (−0.866 − 0.5i)37-s + (−0.173 − 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.766 − 0.642i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.998 - 0.0622i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.998 - 0.0622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05865014515 - 1.881406784i\)
\(L(\frac12)\) \(\approx\) \(0.05865014515 - 1.881406784i\)
\(L(1)\) \(\approx\) \(0.9721324301 - 0.4726660444i\)
\(L(1)\) \(\approx\) \(0.9721324301 - 0.4726660444i\)

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 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.984 + 0.173i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
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 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (0.984 + 0.173i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−19.310698038436150394164640214367, −18.447947932088828603184279237913, −17.77071655930485732670058240957, −17.360388037386381337852483152824, −16.44892244297209425919632099885, −15.59547548995596691470885451734, −14.933701879863285230999525065132, −14.366894629463571268098937477719, −13.78916741087487711496488380453, −12.79725702484941283646145563296, −12.14592905133480363215593240762, −11.53176426215747011440786959465, −10.452974022545971733432943410926, −9.93994803952434794265747904453, −9.6433658423534222398297986807, −8.243240254952799535266324356917, −7.655754206181791180407545789685, −6.98683881196400422587934822322, −6.21534564766928196976962615735, −5.36609813587183014745158183002, −4.67036101579441941525546544593, −3.552192401880588971805881568695, −2.88046018274742420605553340884, −2.06851322093411238629073542523, −1.18540240844566676406000135486, 0.36306360391957481651917081677, 0.81774960956008696320094026814, 2.08261120543871046654574895596, 2.76498972864777917258494363342, 3.81849541076770537227779126224, 4.76430516668946965283942568841, 5.2820291777705859343348762322, 6.05268333065464175135511161118, 6.960083683933901977886091388947, 7.83803758553694619065166274328, 8.59295236810753991690001738631, 9.09543482912716002782572010298, 10.055843444211594650211507883958, 10.52356452900794085426303696293, 11.65250633505210920411300716171, 12.21365569573740356781660187973, 12.82681951854342589825088820528, 13.756643206210341778541474693613, 14.068992374821860085174671556306, 15.103401663036032140252897559994, 15.86747072670505650375754628566, 16.53688180396464662845795799193, 17.04261491586156883649016718040, 17.71895078065945696269073759634, 18.56314985987324682688512415020

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