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Properties

Label 1-2040-2040.509-r1-0-0
Degree $1$
Conductor $2040$
Sign $1$
Analytic cond. $219.228$
Root an. cond. $219.228$
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  + 7-s − 11-s + 13-s − 19-s − 23-s − 29-s − 31-s − 37-s + 41-s + 43-s + 47-s + 49-s − 53-s + 59-s + 61-s + 67-s + 71-s + 73-s − 77-s − 79-s − 83-s − 89-s + 91-s + 97-s + ⋯
L(s)  = 1  + 7-s − 11-s + 13-s − 19-s − 23-s − 29-s − 31-s − 37-s + 41-s + 43-s + 47-s + 49-s − 53-s + 59-s + 61-s + 67-s + 71-s + 73-s − 77-s − 79-s − 83-s − 89-s + 91-s + 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 17\)
Sign: $1$
Analytic conductor: \(219.228\)
Root analytic conductor: \(219.228\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2040} (509, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2040,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.099220895\)
\(L(\frac12)\) \(\approx\) \(2.099220895\)
\(L(1)\) \(\approx\) \(1.112896487\)
\(L(1)\) \(\approx\) \(1.112896487\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
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

−19.79728845808479748038056350848, −18.77005224337424979768856280749, −18.34409398930877714275335560191, −17.607931790930677671500570646090, −16.92562219523258552563118047956, −15.90129855472414364063610220147, −15.51279381074521151248484833708, −14.49010164486031359479267278944, −14.016589393729815413062942281690, −13.02748488745455871600151698805, −12.52610560445365186689918836760, −11.37133199332033970571294910975, −10.93317404374383316773541707672, −10.26011702102553166943654146186, −9.164445542883076389517353123, −8.38874138859366822700007968976, −7.844749792353295007533431907850, −6.99495202910249094580266385638, −5.856837744068214005057675995249, −5.37902646692304319541662231580, −4.30806265481465329781411420635, −3.67621488801762355151614213214, −2.37303214950555537705010813021, −1.773300218086715375279264539287, −0.576524590722545700736827359463, 0.576524590722545700736827359463, 1.773300218086715375279264539287, 2.37303214950555537705010813021, 3.67621488801762355151614213214, 4.30806265481465329781411420635, 5.37902646692304319541662231580, 5.856837744068214005057675995249, 6.99495202910249094580266385638, 7.844749792353295007533431907850, 8.38874138859366822700007968976, 9.164445542883076389517353123, 10.26011702102553166943654146186, 10.93317404374383316773541707672, 11.37133199332033970571294910975, 12.52610560445365186689918836760, 13.02748488745455871600151698805, 14.016589393729815413062942281690, 14.49010164486031359479267278944, 15.51279381074521151248484833708, 15.90129855472414364063610220147, 16.92562219523258552563118047956, 17.607931790930677671500570646090, 18.34409398930877714275335560191, 18.77005224337424979768856280749, 19.79728845808479748038056350848

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