Properties

Label 1-3024-3024.1157-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.700 + 0.713i$
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  + (0.984 − 0.173i)5-s + (−0.984 − 0.173i)11-s + (−0.642 − 0.766i)13-s − 17-s i·19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.642 + 0.766i)29-s + (−0.939 − 0.342i)31-s + (−0.866 − 0.5i)37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.939 − 0.342i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)5-s + (−0.984 − 0.173i)11-s + (−0.642 − 0.766i)13-s − 17-s i·19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.642 + 0.766i)29-s + (−0.939 − 0.342i)31-s + (−0.866 − 0.5i)37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.939 − 0.342i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.700 + 0.713i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.700 + 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06694822829 - 0.1595146799i\)
\(L(\frac12)\) \(\approx\) \(-0.06694822829 - 0.1595146799i\)
\(L(1)\) \(\approx\) \(0.9653006398 - 0.1925125694i\)
\(L(1)\) \(\approx\) \(0.9653006398 - 0.1925125694i\)

Euler product

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

−19.16415799244258848511849147907, −18.56524932101202776783084689637, −17.92366477056526130376538124435, −17.238998558531095876201926916742, −16.66026553809113077581380060, −15.83072325183501028038468008625, −15.02421043054009728838802672765, −14.412987686807743553739411406879, −13.62838866925052914478075751072, −13.078380610172635483840627087679, −12.42001832784018389199960422972, −11.40451285997220415652701563596, −10.77844680135148719391198700939, −9.97957569790137683302033456003, −9.43887114554449429872791120377, −8.708576748103207876054344285779, −7.70028214426384628288636195123, −7.0318970642203893858605749957, −6.246890431569681260239394246946, −5.4431739621481573450675723799, −4.84356291105658309318537726094, −3.86316971414535626624188119961, −2.78366525223240058633793956206, −2.13378650165966751496121034412, −1.39984183381420601558962103619, 0.02902237527237565109944346093, 0.821948770470919919420529661735, 2.19611477342701740026583267492, 2.47577296116301318534750571608, 3.53504170917126765184613417601, 4.75821762657264825604251155439, 5.25980418913426741957884717113, 5.89949248371368493135892522772, 6.98513814458638209330876264882, 7.41179207792754367506961835660, 8.69443603333521138451472062132, 8.97578967942137306688122583253, 9.929887891169459854811843383493, 10.72271681521774035151133014534, 11.002214832706727002012634593252, 12.36337246083556207708947415367, 12.876379917729654284990459011332, 13.42173216237409128575211830467, 14.126451577222868729951288042259, 15.05337962995236053706031044420, 15.52546160059000659264521088442, 16.45232796862186811561213007204, 17.14641614994128860334377240427, 17.75233105696634956457988150684, 18.289520806572991671342331966

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