Properties

Label 2-5520-1.1-c1-0-13
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 3·7-s + 9-s + 2.87·11-s + 4.87·13-s − 15-s − 3.87·17-s − 4.87·19-s + 3·21-s + 23-s + 25-s − 27-s + 1.87·29-s − 3·31-s − 2.87·33-s − 3·35-s + 37-s − 4.87·39-s + 1.87·41-s + 11.7·43-s + 45-s − 0.872·47-s + 2·49-s + 3.87·51-s + 3.87·53-s + 2.87·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.13·7-s + 0.333·9-s + 0.866·11-s + 1.35·13-s − 0.258·15-s − 0.939·17-s − 1.11·19-s + 0.654·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.347·29-s − 0.538·31-s − 0.500·33-s − 0.507·35-s + 0.164·37-s − 0.780·39-s + 0.292·41-s + 1.79·43-s + 0.149·45-s − 0.127·47-s + 0.285·49-s + 0.542·51-s + 0.531·53-s + 0.387·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.477975904\)
\(L(\frac12)\) \(\approx\) \(1.477975904\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 2.87T + 11T^{2} \)
13 \( 1 - 4.87T + 13T^{2} \)
17 \( 1 + 3.87T + 17T^{2} \)
19 \( 1 + 4.87T + 19T^{2} \)
29 \( 1 - 1.87T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 1.87T + 41T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + 0.872T + 47T^{2} \)
53 \( 1 - 3.87T + 53T^{2} \)
59 \( 1 - 1.87T + 59T^{2} \)
61 \( 1 - 1.12T + 61T^{2} \)
67 \( 1 - 4.74T + 67T^{2} \)
71 \( 1 + 9.61T + 71T^{2} \)
73 \( 1 - 4.87T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 7.87T + 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−8.297933909585831816790910274458, −7.14018342556354485140555949463, −6.48731721168165193088041337216, −6.21260844886013901889749400283, −5.49580653896185110172653151757, −4.30352876015498437246279401106, −3.87443312575720363457924189693, −2.82673822831659638447813781400, −1.77986557307551806571387187010, −0.67658412356323317084238697296, 0.67658412356323317084238697296, 1.77986557307551806571387187010, 2.82673822831659638447813781400, 3.87443312575720363457924189693, 4.30352876015498437246279401106, 5.49580653896185110172653151757, 6.21260844886013901889749400283, 6.48731721168165193088041337216, 7.14018342556354485140555949463, 8.297933909585831816790910274458

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