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Properties

Label 1-2280-2280.1667-r0-0-0
Degree $1$
Conductor $2280$
Sign $-0.149 + 0.988i$
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  + (0.866 + 0.5i)7-s + (0.5 + 0.866i)11-s + (0.342 + 0.939i)13-s + (−0.642 − 0.766i)17-s + (−0.984 − 0.173i)23-s + (0.766 + 0.642i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (−0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (−0.642 + 0.766i)47-s + (0.5 + 0.866i)49-s + (0.984 + 0.173i)53-s + (−0.766 + 0.642i)59-s + (−0.173 + 0.984i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (0.5 + 0.866i)11-s + (0.342 + 0.939i)13-s + (−0.642 − 0.766i)17-s + (−0.984 − 0.173i)23-s + (0.766 + 0.642i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (−0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (−0.642 + 0.766i)47-s + (0.5 + 0.866i)49-s + (0.984 + 0.173i)53-s + (−0.766 + 0.642i)59-s + (−0.173 + 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.149 + 0.988i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ -0.149 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.028154709 + 1.195683836i\)
\(L(\frac12)\) \(\approx\) \(1.028154709 + 1.195683836i\)
\(L(1)\) \(\approx\) \(1.099024969 + 0.2995002100i\)
\(L(1)\) \(\approx\) \(1.099024969 + 0.2995002100i\)

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 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.342 + 0.939i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.642 + 0.766i)T \)
53 \( 1 + (0.984 + 0.173i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.342 + 0.939i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.642 - 0.766i)T \)
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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

−19.62159127478621236663494672273, −18.70229669268133045150902058135, −17.88046380862999000967841387340, −17.42304287175918692228122996258, −16.66241204093156301359779721307, −15.857797709566523315497522686969, −15.098951442315302125140607482856, −14.39028702842769738518921929829, −13.67284055676775986402603203455, −13.089596743566866087344501851147, −12.095202684411462070813261739969, −11.326274164368989822182512594286, −10.76849070675780510764585804862, −10.06142503775022459161683596180, −9.02644415938931615284497817200, −8.19023700956138291972343529433, −7.8557518997934490370324709902, −6.69555743454802016778885763035, −5.95288318744500283738545251969, −5.21261595819541705786484430650, −4.10698056136195036961152586265, −3.67252997088270239369994860665, −2.44267024457894437682488003810, −1.52474853647890398484044334785, −0.5204742221067958907005401299, 1.35119907335970363170247594408, 1.96959643710795067399961704384, 2.911718396837080988183951256612, 4.17265083480129783843871308723, 4.64219499066508263846666564568, 5.51075552156792117485838048827, 6.55803445899442526947849257548, 7.100040771722345638462697484, 8.0947308568129269772088730603, 8.85713404682640836874318278173, 9.40356536427719914104630205210, 10.39357662574644223492179813465, 11.17582259389477994242428837565, 12.00438587385796334163057327287, 12.25618488473630889274048305176, 13.5304819507263458486249138270, 14.118093277614492719481945962339, 14.74950113882527186520310632380, 15.5418753639453483366629595816, 16.20811937309761807817716379203, 17.04427352339620091971972108073, 17.93912103445456651671139208487, 18.172038153375522770254988057492, 19.10063464634782800233661775891, 19.997224069631728417825268392065

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