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Properties

Label 1-6384-6384.4421-r0-0-0
Degree $1$
Conductor $6384$
Sign $0.556 - 0.830i$
Analytic cond. $29.6471$
Root an. cond. $29.6471$
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 − i·11-s + (0.642 + 0.766i)13-s + (0.939 − 0.342i)17-s + (0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (0.984 − 0.173i)29-s + (0.5 − 0.866i)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 + (−0.984 + 0.173i)53-s + (−0.173 + 0.984i)55-s + (−0.342 − 0.939i)59-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)5-s − i·11-s + (0.642 + 0.766i)13-s + (0.939 − 0.342i)17-s + (0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (0.984 − 0.173i)29-s + (0.5 − 0.866i)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 + (−0.984 + 0.173i)53-s + (−0.173 + 0.984i)55-s + (−0.342 − 0.939i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6384\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.556 - 0.830i$
Analytic conductor: \(29.6471\)
Root analytic conductor: \(29.6471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6384} (4421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6384,\ (0:\ ),\ 0.556 - 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.462140325 - 0.7801942553i\)
\(L(\frac12)\) \(\approx\) \(1.462140325 - 0.7801942553i\)
\(L(1)\) \(\approx\) \(1.011745294 - 0.1572129498i\)
\(L(1)\) \(\approx\) \(1.011745294 - 0.1572129498i\)

Euler product

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

−17.96140153687661869688119054474, −16.955383284094044982069384087712, −16.40062994588271793492169805366, −15.56567083893767664602487821973, −15.223250593953587323089596381714, −14.63333375990882846405403739490, −13.82468582474277472187146006527, −13.01452451719799532067874451742, −12.346297795977560422131705534676, −11.97904357417837522002414899505, −11.048598565992540297837217153274, −10.57516173504817047134138137880, −9.8432507858685457408731402261, −9.0836807300146914357765289536, −8.11344995863137608255281731845, −7.9000453090036320947263596554, −7.0270665076616598085673522208, −6.44214451413306837119863396153, −5.46900134535313342936864162041, −4.78287883082557324238935767654, −4.08956345004920239390214188114, −3.24682962315173024236838324818, −2.826585689632739196993424841858, −1.532243809562216880143864621019, −0.863213666397516738694208682074, 0.59679203075880184256121467807, 1.16870014498054614407652339327, 2.41443861358874561196418702789, 3.2487248911620359715941926339, 3.76931000490613298335505762780, 4.585555413096241797879118345771, 5.22614980047788836530544382319, 6.21274200209819930160010598474, 6.70268878758377887555522908673, 7.64377228794236905110090218265, 8.19189519870745932976850355270, 8.7725163375532700152927893108, 9.44039452031512024745282514013, 10.39233791663695490564622722856, 11.05513898145465892856570136906, 11.59962779425397344658169993618, 12.15002913078597703230649014280, 12.864443138972833446100829752793, 13.743814278629911730219720468704, 14.13046746063773618840827816966, 14.99059279228035770081729568066, 15.67089624493083902694064054372, 16.16109255758950030198472685933, 16.80125144264204281318316634373, 17.25244492102289413482509261569

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