Properties

Label 2-5520-92.91-c1-0-5
Degree $2$
Conductor $5520$
Sign $-0.0331 - 0.999i$
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 − 2.01·7-s − 9-s − 0.697·11-s + 0.195·13-s − 15-s − 0.430i·17-s + 7.71·19-s + 2.01i·21-s + (−4.23 − 2.25i)23-s − 25-s + i·27-s − 9.64·29-s + 1.05i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447i·5-s − 0.762·7-s − 0.333·9-s − 0.210·11-s + 0.0541·13-s − 0.258·15-s − 0.104i·17-s + 1.76·19-s + 0.440i·21-s + (−0.882 − 0.471i)23-s − 0.200·25-s + 0.192i·27-s − 1.79·29-s + 0.188i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0331 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0331 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.0331 - 0.999i$
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: \((2,\ 5520,\ (\ :1/2),\ -0.0331 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4054519114\)
\(L(\frac12)\) \(\approx\) \(0.4054519114\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.23 + 2.25i)T \)
good7 \( 1 + 2.01T + 7T^{2} \)
11 \( 1 + 0.697T + 11T^{2} \)
13 \( 1 - 0.195T + 13T^{2} \)
17 \( 1 + 0.430iT - 17T^{2} \)
19 \( 1 - 7.71T + 19T^{2} \)
29 \( 1 + 9.64T + 29T^{2} \)
31 \( 1 - 1.05iT - 31T^{2} \)
37 \( 1 + 4.89iT - 37T^{2} \)
41 \( 1 + 3.71T + 41T^{2} \)
43 \( 1 - 5.85T + 43T^{2} \)
47 \( 1 - 10.6iT - 47T^{2} \)
53 \( 1 + 0.464iT - 53T^{2} \)
59 \( 1 + 7.64iT - 59T^{2} \)
61 \( 1 - 4.18iT - 61T^{2} \)
67 \( 1 + 3.70T + 67T^{2} \)
71 \( 1 - 5.69iT - 71T^{2} \)
73 \( 1 + 3.51T + 73T^{2} \)
79 \( 1 + 9.68T + 79T^{2} \)
83 \( 1 - 0.982T + 83T^{2} \)
89 \( 1 - 5.48iT - 89T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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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

−8.223724519638295101073941123199, −7.52368187051713912736984297724, −7.09659455428897706285048248460, −6.03371574356254454676762737723, −5.68276511565751572703073101682, −4.77099473625506368562703784751, −3.76190224491722862938866316522, −3.05393292388646243073624655367, −2.06932319816433921565611163474, −1.02841890830234359759569170603, 0.11520398481811549750772374118, 1.67685879129451693656042934741, 2.84563107140758668906852781421, 3.45826220112957325599044337188, 4.09519411836529927255084939144, 5.23971248930784967935490798778, 5.70799462414771539121022071520, 6.49984810839073432120757388551, 7.36114321125421552470308755867, 7.81693597996361481822943673800

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