Properties

Label 2-5520-92.91-c1-0-34
Degree $2$
Conductor $5520$
Sign $0.999 + 0.0202i$
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 − 4.84·7-s − 9-s − 3.09·11-s − 4.53·13-s − 15-s + 1.04i·17-s − 3.05·19-s − 4.84i·21-s + (−4.79 − 0.0971i)23-s − 25-s i·27-s − 2.44·29-s − 3.08i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.447i·5-s − 1.83·7-s − 0.333·9-s − 0.933·11-s − 1.25·13-s − 0.258·15-s + 0.254i·17-s − 0.701·19-s − 1.05i·21-s + (−0.999 − 0.0202i)23-s − 0.200·25-s − 0.192i·27-s − 0.454·29-s − 0.553i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.999 + 0.0202i$
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: \((2,\ 5520,\ (\ :1/2),\ 0.999 + 0.0202i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3359200672\)
\(L(\frac12)\) \(\approx\) \(0.3359200672\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.79 + 0.0971i)T \)
good7 \( 1 + 4.84T + 7T^{2} \)
11 \( 1 + 3.09T + 11T^{2} \)
13 \( 1 + 4.53T + 13T^{2} \)
17 \( 1 - 1.04iT - 17T^{2} \)
19 \( 1 + 3.05T + 19T^{2} \)
29 \( 1 + 2.44T + 29T^{2} \)
31 \( 1 + 3.08iT - 31T^{2} \)
37 \( 1 - 3.78iT - 37T^{2} \)
41 \( 1 - 4.04T + 41T^{2} \)
43 \( 1 + 2.53T + 43T^{2} \)
47 \( 1 - 3.83iT - 47T^{2} \)
53 \( 1 - 8.14iT - 53T^{2} \)
59 \( 1 + 4.64iT - 59T^{2} \)
61 \( 1 + 11.3iT - 61T^{2} \)
67 \( 1 + 3.77T + 67T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 - 2.53T + 73T^{2} \)
79 \( 1 + 1.04T + 79T^{2} \)
83 \( 1 - 4.24T + 83T^{2} \)
89 \( 1 - 3.89iT - 89T^{2} \)
97 \( 1 + 5.72iT - 97T^{2} \)
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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

−8.054860489390846682295094605323, −7.45250601522713560415913406055, −6.59455563731878386568381886175, −6.08557298974710151765757432235, −5.32722936830464359138329705220, −4.37099290393586340582906358470, −3.62843166964713083813964786129, −2.82399788733433908982743417970, −2.28359755793792857030734355096, −0.19028270674210585963516531558, 0.42317292055213164144178485279, 2.05995542978126612898410931975, 2.71825895509536087166495051521, 3.55150235848560526592639487649, 4.51252272666447774523370465870, 5.45699033961699352037130263300, 6.01187099520731621902519535395, 6.82373614727070702727241329193, 7.34552813469010175772294755807, 8.059161675882536485470616067535

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