Properties

Label 2-5586-1.1-c1-0-88
Degree $2$
Conductor $5586$
Sign $1$
Analytic cond. $44.6044$
Root an. cond. $6.67865$
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  + 2-s + 3-s + 4-s + 3.23·5-s + 6-s + 8-s + 9-s + 3.23·10-s − 0.763·11-s + 12-s + 3.23·13-s + 3.23·15-s + 16-s + 7.23·17-s + 18-s − 19-s + 3.23·20-s − 0.763·22-s + 0.763·23-s + 24-s + 5.47·25-s + 3.23·26-s + 27-s − 4.47·29-s + 3.23·30-s − 6.47·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.44·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.02·10-s − 0.230·11-s + 0.288·12-s + 0.897·13-s + 0.835·15-s + 0.250·16-s + 1.75·17-s + 0.235·18-s − 0.229·19-s + 0.723·20-s − 0.162·22-s + 0.159·23-s + 0.204·24-s + 1.09·25-s + 0.634·26-s + 0.192·27-s − 0.830·29-s + 0.590·30-s − 1.16·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5586\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(44.6044\)
Root analytic conductor: \(6.67865\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5586,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.838855922\)
\(L(\frac12)\) \(\approx\) \(5.838855922\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 \)
19 \( 1 + T \)
good5 \( 1 - 3.23T + 5T^{2} \)
11 \( 1 + 0.763T + 11T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
17 \( 1 - 7.23T + 17T^{2} \)
23 \( 1 - 0.763T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 6.47T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 + 2.47T + 43T^{2} \)
47 \( 1 + 4.94T + 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 + 6.47T + 61T^{2} \)
67 \( 1 - 4.76T + 67T^{2} \)
71 \( 1 + 6.47T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 1.70T + 79T^{2} \)
83 \( 1 + 5.52T + 83T^{2} \)
89 \( 1 - 8.94T + 89T^{2} \)
97 \( 1 - 5.70T + 97T^{2} \)
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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

−7.949125267703662448943301973953, −7.51341261841521734379512898341, −6.44464366500312635397871447642, −5.89989681469861216576395855619, −5.42160184223049131107804410887, −4.50782068479063759546497239644, −3.50274724647496148946517921456, −2.94178703665131840615315962462, −1.93550869446700512961426831069, −1.29436899002739379186805690245, 1.29436899002739379186805690245, 1.93550869446700512961426831069, 2.94178703665131840615315962462, 3.50274724647496148946517921456, 4.50782068479063759546497239644, 5.42160184223049131107804410887, 5.89989681469861216576395855619, 6.44464366500312635397871447642, 7.51341261841521734379512898341, 7.949125267703662448943301973953

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