Properties

Label 2-5520-92.91-c1-0-22
Degree $2$
Conductor $5520$
Sign $-0.949 - 0.314i$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + i·5-s − 0.482·7-s − 9-s − 1.10·11-s + 2.25·13-s − 15-s + 6.03i·17-s + 1.48·19-s − 0.482i·21-s + (4.55 + 1.50i)23-s − 25-s i·27-s − 7.68·29-s − 5.12i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.447i·5-s − 0.182·7-s − 0.333·9-s − 0.332·11-s + 0.625·13-s − 0.258·15-s + 1.46i·17-s + 0.340·19-s − 0.105i·21-s + (0.949 + 0.314i)23-s − 0.200·25-s − 0.192i·27-s − 1.42·29-s − 0.920i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.949 - 0.314i$
Analytic conductor: \(44.0774\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ -0.949 - 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.204193085\)
\(L(\frac12)\) \(\approx\) \(1.204193085\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 - iT \)
23 \( 1 + (-4.55 - 1.50i)T \)
good7 \( 1 + 0.482T + 7T^{2} \)
11 \( 1 + 1.10T + 11T^{2} \)
13 \( 1 - 2.25T + 13T^{2} \)
17 \( 1 - 6.03iT - 17T^{2} \)
19 \( 1 - 1.48T + 19T^{2} \)
29 \( 1 + 7.68T + 29T^{2} \)
31 \( 1 + 5.12iT - 31T^{2} \)
37 \( 1 - 5.83iT - 37T^{2} \)
41 \( 1 - 1.27T + 41T^{2} \)
43 \( 1 - 8.46T + 43T^{2} \)
47 \( 1 - 11.2iT - 47T^{2} \)
53 \( 1 + 8.81iT - 53T^{2} \)
59 \( 1 - 8.08iT - 59T^{2} \)
61 \( 1 + 0.0392iT - 61T^{2} \)
67 \( 1 - 8.39T + 67T^{2} \)
71 \( 1 + 1.25iT - 71T^{2} \)
73 \( 1 + 3.47T + 73T^{2} \)
79 \( 1 - 3.61T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 + 6.85iT - 89T^{2} \)
97 \( 1 + 0.330iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.440914321235256399779489952641, −7.82882003720763578040832251534, −7.07766627562421600282893265681, −6.11794399938805080794164923522, −5.76813961667908951787597875886, −4.78865541003802237759281076723, −3.91691114296765137363249907189, −3.36820908662381538237592777439, −2.44592612954395511073839006339, −1.30346048427862691993128791674, 0.33103828136453240524683841560, 1.30447436318360766801696418597, 2.39550749308442522568449245317, 3.20190101526348462244816435583, 4.10893437493273155662325172291, 5.19004186333119214138433743852, 5.50217175417383166772544756946, 6.53857686975647509519586094621, 7.17413715094601519925375265835, 7.72176555234247378839431194623

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