Properties

Label 2-5586-1.1-c1-0-110
Degree $2$
Conductor $5586$
Sign $-1$
Analytic cond. $44.6044$
Root an. cond. $6.67865$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 2·5-s + 6-s + 8-s + 9-s − 2·10-s − 4·11-s + 12-s − 2·13-s − 2·15-s + 16-s + 6·17-s + 18-s + 19-s − 2·20-s − 4·22-s − 4·23-s + 24-s − 25-s − 2·26-s + 27-s − 2·29-s − 2·30-s − 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s − 0.852·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.371·29-s − 0.365·30-s − 0.718·31-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: yes
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.70941961691065571239874962190, −7.43853332815428300125548400148, −6.24339547721559801046855566948, −5.51603398775018548116922398014, −4.78038792313611077738249080672, −4.02809177468296829616623514392, −3.26227108565245156502288611307, −2.68432652996673587498769750331, −1.58036555306778050959208119703, 0, 1.58036555306778050959208119703, 2.68432652996673587498769750331, 3.26227108565245156502288611307, 4.02809177468296829616623514392, 4.78038792313611077738249080672, 5.51603398775018548116922398014, 6.24339547721559801046855566948, 7.43853332815428300125548400148, 7.70941961691065571239874962190

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