Properties

Label 2-5586-1.1-c1-0-15
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 + 2.49·5-s + 6-s − 8-s + 9-s − 2.49·10-s − 3.46·11-s − 12-s − 4.98·13-s − 2.49·15-s + 16-s + 2.24·17-s − 18-s + 19-s + 2.49·20-s + 3.46·22-s + 6.00·23-s + 24-s + 1.21·25-s + 4.98·26-s − 27-s − 5.51·29-s + 2.49·30-s − 7.28·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.11·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.788·10-s − 1.04·11-s − 0.288·12-s − 1.38·13-s − 0.643·15-s + 0.250·16-s + 0.544·17-s − 0.235·18-s + 0.229·19-s + 0.557·20-s + 0.738·22-s + 1.25·23-s + 0.204·24-s + 0.242·25-s + 0.977·26-s − 0.192·27-s − 1.02·29-s + 0.455·30-s − 1.30·31-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: $\chi_{5586} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5586,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.050059500\)
\(L(\frac12)\) \(\approx\) \(1.050059500\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.49T + 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 4.98T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
23 \( 1 - 6.00T + 23T^{2} \)
29 \( 1 + 5.51T + 29T^{2} \)
31 \( 1 + 7.28T + 31T^{2} \)
37 \( 1 + 9.16T + 37T^{2} \)
41 \( 1 - 4.64T + 41T^{2} \)
43 \( 1 - 9.54T + 43T^{2} \)
47 \( 1 - 5.36T + 47T^{2} \)
53 \( 1 + 5.87T + 53T^{2} \)
59 \( 1 + 8.67T + 59T^{2} \)
61 \( 1 - 4.37T + 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 - 14.4T + 73T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 + 5.59T + 83T^{2} \)
89 \( 1 - 7.52T + 89T^{2} \)
97 \( 1 - 7.00T + 97T^{2} \)
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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

−8.020758640646063974376171629995, −7.35520867859197461360649381908, −6.93713274929115785005196719710, −5.85059592794873442722624307545, −5.41460955225856559116393703934, −4.87453076045653853893701419983, −3.48866677007819485940150159308, −2.46452221540714075445432646705, −1.87325107224201813687647184864, −0.60887240030285224411112307281, 0.60887240030285224411112307281, 1.87325107224201813687647184864, 2.46452221540714075445432646705, 3.48866677007819485940150159308, 4.87453076045653853893701419983, 5.41460955225856559116393703934, 5.85059592794873442722624307545, 6.93713274929115785005196719710, 7.35520867859197461360649381908, 8.020758640646063974376171629995

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