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Properties

Label 1-2280-2280.749-r0-0-0
Degree $1$
Conductor $2280$
Sign $-0.813 - 0.582i$
Analytic cond. $10.5882$
Root an. cond. $10.5882$
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  − 7-s + 11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)29-s − 31-s − 37-s + (−0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + 49-s + (−0.5 + 0.866i)53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯
L(s)  = 1  − 7-s + 11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)29-s − 31-s − 37-s + (−0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + 49-s + (−0.5 + 0.866i)53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.813 - 0.582i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ -0.813 - 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1883896487 - 0.5866099898i\)
\(L(\frac12)\) \(\approx\) \(0.1883896487 - 0.5866099898i\)
\(L(1)\) \(\approx\) \(0.8382728957 - 0.1345546494i\)
\(L(1)\) \(\approx\) \(0.8382728957 - 0.1345546494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−19.7943599121690934842747115756, −19.34954503852213806724555016010, −18.5765903009886855560888101118, −17.83791773628358410098451972717, −16.901184728382359745730244928257, −16.38708826137342794625232150093, −15.81444261301880719115963442893, −14.74743202157448921841929437967, −14.29502153591552259872799789577, −13.31912788838181279182017062449, −12.76587992547538900281184487306, −11.95272086859537930751714392724, −11.251568173846399073655916039769, −10.350691300470218179760574706314, −9.67020281372597550892673637790, −8.82382153384330197086835600762, −8.38737671836556086974513131154, −6.94908214110789865455122965839, −6.62687292187893340294051655050, −5.924480159059686759754154439772, −4.76247331726266862120370261623, −3.86917629074067141456896155646, −3.36042857532573239710311629235, −2.1290020563944734249545028236, −1.31391722987304829810979057224, 0.206009326872081960684145349842, 1.38081419874806367653411175066, 2.49154529835799285170252199395, 3.449873712316334854409168988, 3.94204053904506519019508840561, 5.14916939855492468084278454140, 5.92515259311962417674664710331, 6.67335728085213826144359629094, 7.33695713831051951126631599509, 8.352166945467234440183768127184, 9.16523482021614209880002075160, 9.7155548003313255025083208221, 10.53834191901975622990659730535, 11.39222567742206992316170883138, 12.13380570175245590183822282695, 12.81771590562367997616828943955, 13.68914385498217792378057739089, 14.07624767549158040608371807063, 15.37180668579951905897331887885, 15.61881305255770456290529325821, 16.486312778414555463829909007650, 17.22938364511043687023162939973, 17.90200578237478504332657682179, 18.71365656236329836549237871511, 19.44957476422950629473131402399

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