Properties

Label 2-5586-1.1-c1-0-92
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 1.58·5-s − 6-s + 8-s + 9-s − 1.58·10-s − 11-s − 12-s + 2.82·13-s + 1.58·15-s + 16-s − 4.24·17-s + 18-s + 19-s − 1.58·20-s − 22-s + 3.41·23-s − 24-s − 2.48·25-s + 2.82·26-s − 27-s + 2.17·29-s + 1.58·30-s − 10.6·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.709·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.501·10-s − 0.301·11-s − 0.288·12-s + 0.784·13-s + 0.409·15-s + 0.250·16-s − 1.02·17-s + 0.235·18-s + 0.229·19-s − 0.354·20-s − 0.213·22-s + 0.711·23-s − 0.204·24-s − 0.497·25-s + 0.554·26-s − 0.192·27-s + 0.403·29-s + 0.289·30-s − 1.91·31-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: \(1\)
Selberg data: \((2,\ 5586,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
19 \( 1 - T \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
23 \( 1 - 3.41T + 23T^{2} \)
29 \( 1 - 2.17T + 29T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
37 \( 1 - 0.242T + 37T^{2} \)
41 \( 1 + 0.585T + 41T^{2} \)
43 \( 1 - 6.24T + 43T^{2} \)
47 \( 1 - 3.17T + 47T^{2} \)
53 \( 1 - 1.82T + 53T^{2} \)
59 \( 1 - 6.07T + 59T^{2} \)
61 \( 1 + 4.82T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 7.89T + 71T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 + 0.171T + 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 - 3.75T + 89T^{2} \)
97 \( 1 + 5.58T + 97T^{2} \)
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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

−7.38556353336312878038613447465, −7.20376195431669540433880207836, −6.15748150583024484928937796812, −5.67123041018279430921433965550, −4.80805777510636754975723185551, −4.13543208222462487273065440528, −3.50183375947912428865665553050, −2.48467441877462232304901553534, −1.35857257526470938939943373715, 0, 1.35857257526470938939943373715, 2.48467441877462232304901553534, 3.50183375947912428865665553050, 4.13543208222462487273065440528, 4.80805777510636754975723185551, 5.67123041018279430921433965550, 6.15748150583024484928937796812, 7.20376195431669540433880207836, 7.38556353336312878038613447465

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