Properties

Label 2-5520-92.91-c1-0-2
Degree $2$
Conductor $5520$
Sign $-0.272 + 0.962i$
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 − 3.74·7-s − 9-s + 0.951·11-s − 0.958·13-s − 15-s + 7.12i·17-s − 8.14·19-s − 3.74i·21-s + (3.34 + 3.43i)23-s − 25-s i·27-s + 3.97·29-s + 10.2i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.447i·5-s − 1.41·7-s − 0.333·9-s + 0.286·11-s − 0.265·13-s − 0.258·15-s + 1.72i·17-s − 1.86·19-s − 0.818i·21-s + (0.697 + 0.716i)23-s − 0.200·25-s − 0.192i·27-s + 0.738·29-s + 1.84i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.272 + 0.962i$
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: \((2,\ 5520,\ (\ :1/2),\ -0.272 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1841674883\)
\(L(\frac12)\) \(\approx\) \(0.1841674883\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.34 - 3.43i)T \)
good7 \( 1 + 3.74T + 7T^{2} \)
11 \( 1 - 0.951T + 11T^{2} \)
13 \( 1 + 0.958T + 13T^{2} \)
17 \( 1 - 7.12iT - 17T^{2} \)
19 \( 1 + 8.14T + 19T^{2} \)
29 \( 1 - 3.97T + 29T^{2} \)
31 \( 1 - 10.2iT - 31T^{2} \)
37 \( 1 - 7.97iT - 37T^{2} \)
41 \( 1 - 2.05T + 41T^{2} \)
43 \( 1 - 4.47T + 43T^{2} \)
47 \( 1 + 5.73iT - 47T^{2} \)
53 \( 1 + 5.84iT - 53T^{2} \)
59 \( 1 + 5.24iT - 59T^{2} \)
61 \( 1 + 1.55iT - 61T^{2} \)
67 \( 1 + 8.31T + 67T^{2} \)
71 \( 1 + 14.9iT - 71T^{2} \)
73 \( 1 - 2.66T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 - 3.86T + 83T^{2} \)
89 \( 1 - 3.08iT - 89T^{2} \)
97 \( 1 - 5.79iT - 97T^{2} \)
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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

−8.703485770548908919861219920415, −8.129220338231806630047037860346, −6.94636173425583588035179915036, −6.48988734618097643952123094422, −6.01465969513600981625198541863, −4.97383712472306811075314021200, −4.09301622369444806412025894866, −3.45408667026583160072483841478, −2.80502801055669301654084130976, −1.64824236203600840153744239470, 0.05854945948714294023653172970, 0.867385685733382361409051102809, 2.42687420021126207148708566332, 2.76322126942181254514241247416, 4.02947940886559851186192022925, 4.59645564781851457575602369420, 5.75076402311436873191600802579, 6.22667678988638572632419432262, 6.97887920563812808238913337597, 7.45614678064388608904870433512

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