Properties

Label 2-5610-1.1-c1-0-86
Degree $2$
Conductor $5610$
Sign $-1$
Analytic cond. $44.7960$
Root an. cond. $6.69298$
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 − 5-s − 6-s − 8-s + 9-s + 10-s − 11-s + 12-s + 4·13-s − 15-s + 16-s + 17-s − 18-s − 8·19-s − 20-s + 22-s + 4·23-s − 24-s + 25-s − 4·26-s + 27-s + 30-s − 6·31-s − 32-s − 33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.213·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.182·30-s − 1.07·31-s − 0.176·32-s − 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5610\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(44.7960\)
Root analytic conductor: \(6.69298\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
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

−7.964530061005052935152738706542, −7.18919828505513955554840895885, −6.59768325293724886185267854945, −5.79011187560713246713535949679, −4.79089783636017760791667087573, −3.85608775455149378248994563062, −3.24457224082782795219686982889, −2.22477921795922970719300310054, −1.35762315363705448822049766533, 0, 1.35762315363705448822049766533, 2.22477921795922970719300310054, 3.24457224082782795219686982889, 3.85608775455149378248994563062, 4.79089783636017760791667087573, 5.79011187560713246713535949679, 6.59768325293724886185267854945, 7.18919828505513955554840895885, 7.964530061005052935152738706542

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