Properties

Label 2-5520-92.91-c1-0-84
Degree $2$
Conductor $5520$
Sign $-0.310 + 0.950i$
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 + 1.70·7-s − 9-s + 3.79·11-s + 1.61·13-s − 15-s − 2.05i·17-s − 0.518·19-s − 1.70i·21-s + (−4.69 + 0.988i)23-s − 25-s + i·27-s − 2.88·29-s − 4.16i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447i·5-s + 0.643·7-s − 0.333·9-s + 1.14·11-s + 0.448·13-s − 0.258·15-s − 0.498i·17-s − 0.119·19-s − 0.371i·21-s + (−0.978 + 0.206i)23-s − 0.200·25-s + 0.192i·27-s − 0.535·29-s − 0.748i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.310 + 0.950i$
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: \((2,\ 5520,\ (\ :1/2),\ -0.310 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.045020184\)
\(L(\frac12)\) \(\approx\) \(2.045020184\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.69 - 0.988i)T \)
good7 \( 1 - 1.70T + 7T^{2} \)
11 \( 1 - 3.79T + 11T^{2} \)
13 \( 1 - 1.61T + 13T^{2} \)
17 \( 1 + 2.05iT - 17T^{2} \)
19 \( 1 + 0.518T + 19T^{2} \)
29 \( 1 + 2.88T + 29T^{2} \)
31 \( 1 + 4.16iT - 31T^{2} \)
37 \( 1 + 7.56iT - 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 4.43T + 43T^{2} \)
47 \( 1 + 0.599iT - 47T^{2} \)
53 \( 1 - 2.97iT - 53T^{2} \)
59 \( 1 + 9.67iT - 59T^{2} \)
61 \( 1 - 2.26iT - 61T^{2} \)
67 \( 1 - 9.80T + 67T^{2} \)
71 \( 1 + 9.51iT - 71T^{2} \)
73 \( 1 + 7.31T + 73T^{2} \)
79 \( 1 - 7.33T + 79T^{2} \)
83 \( 1 - 4.63T + 83T^{2} \)
89 \( 1 + 16.0iT - 89T^{2} \)
97 \( 1 - 14.4iT - 97T^{2} \)
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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

−7.83793109686308366454087525600, −7.40947284150059640010862886821, −6.40006247756074555422595987488, −5.93718603647348774969931176917, −5.06523025023818978027453056757, −4.20585416783454357756129667770, −3.57138402560501626586761117882, −2.26409217237648921067088437034, −1.57179600562302942149221833564, −0.56840435121473957590523487527, 1.20374872156259294409560338940, 2.14694920599887478936678827622, 3.26689122278431770145987491542, 3.97845864457296593250648723984, 4.55231290218120486901089577493, 5.52581926154309936954592614402, 6.23840875556294053002987353769, 6.81411488818707607463921740110, 7.79426528662253225138580073006, 8.391816309532432933125597094769

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