Properties

Degree $2$
Conductor $5610$
Sign $1$
Motivic weight $1$
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 − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 14-s + 15-s + 16-s + 17-s − 18-s + 4·19-s − 20-s − 21-s + 22-s + 5·23-s + 24-s + 25-s − 27-s + 28-s + 9·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.213·22-s + 1.04·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s + 1.67·29-s − 0.182·30-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: $1$
Motivic weight: \(1\)
Character: $\chi_{5610} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.026581080\)
\(L(\frac12)\) \(\approx\) \(1.026581080\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−17.57197716405879, −16.97493925903809, −16.41917032399416, −15.90467129748071, −15.31140350499814, −14.73872534482453, −13.95816447055088, −13.24617553814845, −12.44719836468617, −11.89707306246455, −11.45729720474090, −10.56366554676421, −10.44283372517615, −9.392227984664880, −8.868647120225653, −8.066790952160015, −7.448021105695814, −6.929123612826360, −6.070236436894749, −5.242447264927804, −4.680029629599484, −3.565894203036517, −2.763717956271738, −1.553044678621692, −0.6532119708245814, 0.6532119708245814, 1.553044678621692, 2.763717956271738, 3.565894203036517, 4.680029629599484, 5.242447264927804, 6.070236436894749, 6.929123612826360, 7.448021105695814, 8.066790952160015, 8.868647120225653, 9.392227984664880, 10.44283372517615, 10.56366554676421, 11.45729720474090, 11.89707306246455, 12.44719836468617, 13.24617553814845, 13.95816447055088, 14.73872534482453, 15.31140350499814, 15.90467129748071, 16.41917032399416, 16.97493925903809, 17.57197716405879

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