Properties

Label 2-5586-1.1-c1-0-66
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 − 4.41·5-s − 6-s + 8-s + 9-s − 4.41·10-s − 11-s − 12-s − 2.82·13-s + 4.41·15-s + 16-s + 4.24·17-s + 18-s + 19-s − 4.41·20-s − 22-s + 0.585·23-s − 24-s + 14.4·25-s − 2.82·26-s − 27-s + 7.82·29-s + 4.41·30-s + 0.656·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.97·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.39·10-s − 0.301·11-s − 0.288·12-s − 0.784·13-s + 1.13·15-s + 0.250·16-s + 1.02·17-s + 0.235·18-s + 0.229·19-s − 0.987·20-s − 0.213·22-s + 0.122·23-s − 0.204·24-s + 2.89·25-s − 0.554·26-s − 0.192·27-s + 1.45·29-s + 0.805·30-s + 0.117·31-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 \( 1 + 4.41T + 5T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
23 \( 1 - 0.585T + 23T^{2} \)
29 \( 1 - 7.82T + 29T^{2} \)
31 \( 1 - 0.656T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 + 2.24T + 43T^{2} \)
47 \( 1 - 8.82T + 47T^{2} \)
53 \( 1 + 3.82T + 53T^{2} \)
59 \( 1 + 8.07T + 59T^{2} \)
61 \( 1 - 0.828T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 9.17T + 73T^{2} \)
79 \( 1 + 5.82T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + 8.41T + 97T^{2} \)
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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

−7.69455050728402810596011300212, −7.07234083213947470372541896923, −6.47227086400524367552556999968, −5.29816877415669655321696242430, −4.89562429515684703647924779871, −4.14077065403950660365487494320, −3.42342812643037481036938482380, −2.73363027048482320977641619538, −1.14098873665669935040871863935, 0, 1.14098873665669935040871863935, 2.73363027048482320977641619538, 3.42342812643037481036938482380, 4.14077065403950660365487494320, 4.89562429515684703647924779871, 5.29816877415669655321696242430, 6.47227086400524367552556999968, 7.07234083213947470372541896923, 7.69455050728402810596011300212

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