Properties

Label 2-5520-1.1-c1-0-62
Degree $2$
Conductor $5520$
Sign $-1$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 7-s + 9-s − 2.24·11-s − 5.41·13-s − 15-s − 4.41·17-s + 7.07·19-s − 21-s + 23-s + 25-s − 27-s + 1.24·29-s + 10.6·31-s + 2.24·33-s + 35-s − 3·37-s + 5.41·39-s − 1.58·41-s − 2·43-s + 45-s − 5.41·47-s − 6·49-s + 4.41·51-s − 6.41·53-s − 2.24·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s − 0.676·11-s − 1.50·13-s − 0.258·15-s − 1.07·17-s + 1.62·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.230·29-s + 1.91·31-s + 0.390·33-s + 0.169·35-s − 0.493·37-s + 0.866·39-s − 0.247·41-s − 0.304·43-s + 0.149·45-s − 0.789·47-s − 0.857·49-s + 0.618·51-s − 0.881·53-s − 0.302·55-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - T \)
23 \( 1 - T \)
good7 \( 1 - T + 7T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 + 5.41T + 13T^{2} \)
17 \( 1 + 4.41T + 17T^{2} \)
19 \( 1 - 7.07T + 19T^{2} \)
29 \( 1 - 1.24T + 29T^{2} \)
31 \( 1 - 10.6T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 + 1.58T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 5.41T + 47T^{2} \)
53 \( 1 + 6.41T + 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 - 1.07T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 2.07T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 + 5.65T + 79T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 1.17T + 97T^{2} \)
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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

−7.74953599290933879962266125461, −6.99634598576095783817616187693, −6.46213577116685072095103128848, −5.40689523529663712690479257581, −5.01550542870153924814862024251, −4.41599738224735621597318883961, −3.06259212135660160400564661163, −2.38698678545669935520309868642, −1.29673460011563917580190857719, 0, 1.29673460011563917580190857719, 2.38698678545669935520309868642, 3.06259212135660160400564661163, 4.41599738224735621597318883961, 5.01550542870153924814862024251, 5.40689523529663712690479257581, 6.46213577116685072095103128848, 6.99634598576095783817616187693, 7.74953599290933879962266125461

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