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Properties

Label 1-2280-2280.869-r0-0-0
Degree $1$
Conductor $2280$
Sign $-0.392 - 0.919i$
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  + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (0.766 − 0.642i)17-s + (0.173 − 0.984i)23-s + (−0.766 − 0.642i)29-s + (0.5 − 0.866i)31-s − 37-s + (−0.939 − 0.342i)41-s + (0.173 + 0.984i)43-s + (0.766 + 0.642i)47-s + (−0.5 − 0.866i)49-s + (0.173 − 0.984i)53-s + (−0.766 + 0.642i)59-s + (−0.173 + 0.984i)61-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (0.766 − 0.642i)17-s + (0.173 − 0.984i)23-s + (−0.766 − 0.642i)29-s + (0.5 − 0.866i)31-s − 37-s + (−0.939 − 0.342i)41-s + (0.173 + 0.984i)43-s + (0.766 + 0.642i)47-s + (−0.5 − 0.866i)49-s + (0.173 − 0.984i)53-s + (−0.766 + 0.642i)59-s + (−0.173 + 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.392 - 0.919i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (869, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ -0.392 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8524044959 - 1.290316829i\)
\(L(\frac12)\) \(\approx\) \(0.8524044959 - 1.290316829i\)
\(L(1)\) \(\approx\) \(1.055250627 - 0.3640389270i\)
\(L(1)\) \(\approx\) \(1.055250627 - 0.3640389270i\)

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

−19.95101977996430052983467821916, −18.92041854396101932044530284684, −18.552035534647777723158026850616, −17.764284829226892647112998592494, −17.13568867790360749971934124307, −16.21783129502165808263736212863, −15.33593565346316770077710913842, −15.12765092156440655509260557420, −14.05800905664596830663035471685, −13.45241947037299247323703640744, −12.339624323005845251646009223, −12.14219797469130438816780005691, −11.05010535788087236866664215123, −10.4780782287606130237907800082, −9.50740277578373666369972417532, −8.807684073626227276601656331547, −8.09560524388353593646378734197, −7.3132261025385285690396047896, −6.40735760214056426560199687079, −5.46859321860392822859710755310, −5.01932771746564532901576947060, −3.87214486994148215276214050901, −3.112645206158526854232175958673, −1.93876401673770138734672552644, −1.434568972386721036974105004630, 0.52753568298180687529376517862, 1.34168193579519753835501097462, 2.57947625840727943065211826685, 3.44579876260072607235158118256, 4.19772504015522779719347590806, 5.15359857115012853837315805060, 5.88418808795404541769452159187, 6.752593649034918803270804550677, 7.72626810830409758609186766233, 8.17199611084153893873195055786, 9.03636270633999808335963701856, 10.05999587484880622046358766762, 10.69431411937516631223045478345, 11.27550313400548854979444759909, 12.07423471929842888559835958496, 13.14086542670869673435110224066, 13.621548480968187414507950829587, 14.248144848514781479059555239646, 15.1153142872975096256726608662, 15.93026069702715686717710729962, 16.612232154930358691973873565224, 17.1589302303291816561195301468, 18.13974501531887456031917594102, 18.63231781560786828360201307565, 19.35163428278753516595874759526

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