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Properties

Label 1-440-440.219-r0-0-0
Degree $1$
Conductor $440$
Sign $1$
Analytic cond. $2.04335$
Root an. cond. $2.04335$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 13-s + 17-s − 19-s + 21-s + 23-s − 27-s + 29-s − 31-s + 37-s + 39-s − 41-s + 43-s + 47-s + 49-s − 51-s + 53-s + 57-s + 59-s + 61-s − 63-s − 67-s − 69-s − 71-s + 73-s + ⋯
L(s)  = 1  − 3-s − 7-s + 9-s − 13-s + 17-s − 19-s + 21-s + 23-s − 27-s + 29-s − 31-s + 37-s + 39-s − 41-s + 43-s + 47-s + 49-s − 51-s + 53-s + 57-s + 59-s + 61-s − 63-s − 67-s − 69-s − 71-s + 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(2.04335\)
Root analytic conductor: \(2.04335\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{440} (219, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 440,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7502734958\)
\(L(\frac12)\) \(\approx\) \(0.7502734958\)
\(L(1)\) \(\approx\) \(0.7126374360\)
\(L(1)\) \(\approx\) \(0.7126374360\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 - T \)
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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

−23.725816872424116214897402463814, −23.29396858532091997094534944275, −22.328437127028198479067634629326, −21.7570541069268074088745726846, −20.8113001726488819089774506197, −19.4842394361912032209461247805, −18.98492632824841024322330205505, −17.94858791595746089330621524421, −16.89822451554131892735648817716, −16.56441324057650110603415747412, −15.479196093038203844073782028282, −14.59678112680324825256081490528, −13.218211706703162665819275372061, −12.54395498009012034731284520708, −11.83592081677514890175440784461, −10.62729670458200995920935567944, −10.00346635272957889861281380278, −9.03057106121922197044615465727, −7.526452546026192423887387766384, −6.74404982863630990024061286335, −5.82323281720518341046721082239, −4.88631177106453436844260060986, −3.74500949115473159710340299213, −2.43491780205073226844995805797, −0.77652854636310367516441322432, 0.77652854636310367516441322432, 2.43491780205073226844995805797, 3.74500949115473159710340299213, 4.88631177106453436844260060986, 5.82323281720518341046721082239, 6.74404982863630990024061286335, 7.526452546026192423887387766384, 9.03057106121922197044615465727, 10.00346635272957889861281380278, 10.62729670458200995920935567944, 11.83592081677514890175440784461, 12.54395498009012034731284520708, 13.218211706703162665819275372061, 14.59678112680324825256081490528, 15.479196093038203844073782028282, 16.56441324057650110603415747412, 16.89822451554131892735648817716, 17.94858791595746089330621524421, 18.98492632824841024322330205505, 19.4842394361912032209461247805, 20.8113001726488819089774506197, 21.7570541069268074088745726846, 22.328437127028198479067634629326, 23.29396858532091997094534944275, 23.725816872424116214897402463814

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