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Properties

Label 1-5520-5520.5099-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.986 + 0.165i$
Analytic cond. $25.6347$
Root an. cond. $25.6347$
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  + (−0.841 + 0.540i)7-s + (−0.989 − 0.142i)11-s + (−0.540 + 0.841i)13-s + (−0.959 − 0.281i)17-s + (−0.281 − 0.959i)19-s + (0.281 − 0.959i)29-s + (−0.415 − 0.909i)31-s + (−0.755 + 0.654i)37-s + (−0.654 + 0.755i)41-s + (0.909 + 0.415i)43-s − 47-s + (0.415 − 0.909i)49-s + (0.540 + 0.841i)53-s + (−0.540 + 0.841i)59-s + (0.909 − 0.415i)61-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)7-s + (−0.989 − 0.142i)11-s + (−0.540 + 0.841i)13-s + (−0.959 − 0.281i)17-s + (−0.281 − 0.959i)19-s + (0.281 − 0.959i)29-s + (−0.415 − 0.909i)31-s + (−0.755 + 0.654i)37-s + (−0.654 + 0.755i)41-s + (0.909 + 0.415i)43-s − 47-s + (0.415 − 0.909i)49-s + (0.540 + 0.841i)53-s + (−0.540 + 0.841i)59-s + (0.909 − 0.415i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.986 + 0.165i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (5099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.986 + 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7152623312 + 0.05976389323i\)
\(L(\frac12)\) \(\approx\) \(0.7152623312 + 0.05976389323i\)
\(L(1)\) \(\approx\) \(0.7297407507 + 0.03693533857i\)
\(L(1)\) \(\approx\) \(0.7297407507 + 0.03693533857i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (-0.841 + 0.540i)T \)
11 \( 1 + (-0.989 - 0.142i)T \)
13 \( 1 + (-0.540 + 0.841i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (-0.281 - 0.959i)T \)
29 \( 1 + (0.281 - 0.959i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + (-0.755 + 0.654i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 - T \)
53 \( 1 + (0.540 + 0.841i)T \)
59 \( 1 + (-0.540 + 0.841i)T \)
61 \( 1 + (0.909 - 0.415i)T \)
67 \( 1 + (-0.989 + 0.142i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (-0.959 + 0.281i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (0.755 - 0.654i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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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

−17.77956513895941920095095804833, −17.30545987990434468479529378864, −16.3532496829744050717482228863, −15.99565055264619491757425859101, −15.27815933803185685051352038573, −14.58339876393075065931920655596, −13.82097508490779959921475878572, −13.094717676186354750909358577667, −12.641992491817017390345338355950, −12.112468384180427317750526694738, −10.88061013402008384595233726756, −10.47950306049645474740380554519, −10.03391191019997523649569551160, −9.09348270661494027505299236289, −8.42041088825646943745294668682, −7.61785729294234227618067700348, −7.03152320954479237625109160866, −6.33467458795394995322777762692, −5.444201569759068886133007193237, −4.91077546071649473730775400397, −3.87461893202330610888427461092, −3.3048629068313822415715585747, −2.48681570287124569015106849996, −1.6602735589628583725154480678, −0.40495872092198016741905163168, 0.40198142297288678490186089144, 1.85381845998515435676081028091, 2.58230408974359594703428528149, 3.02871709566224119178948944654, 4.23022987011406017909148425238, 4.73729741221925034404685910875, 5.6311118442544246176755156479, 6.338460866679306352501013120513, 6.96437619824020683818862776465, 7.67361437875327812461469939325, 8.62817476782135415716272774212, 9.11452207212659698321291326944, 9.83814882586605688205034288281, 10.43169440975587528689840087931, 11.42921610645918544761156434285, 11.74372188929096385483746465934, 12.7577801019445375592880080875, 13.22915372607833536499809084671, 13.717843110961180039025048967000, 14.7292921105205368124705694535, 15.38512976585104540196892834128, 15.83050191758417784099215262616, 16.48974683073401629000739806172, 17.234347389449735916415447598003, 17.90390598221531850983682082531

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