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 + 2·7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 4·13-s − 2·14-s − 15-s + 16-s − 17-s − 18-s − 6·19-s + 20-s − 2·21-s + 22-s − 6·23-s + 24-s + 25-s + 4·26-s − 27-s + 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.755·7-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 − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.436·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-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: $\chi_{5610} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9199306675\)
\(L(\frac12)\) \(\approx\) \(0.9199306675\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.201869185986356142379883995436, −7.47425803148862222758839823334, −6.73862341612796237656813818385, −6.13069987557207208474693413836, −5.24608275465868938671918711089, −4.71088225378315451728474525386, −3.71887600825931860487631442624, −2.27446841186633416719373384707, −1.97529709676451637081125135565, −0.56506647049289628628130404500, 0.56506647049289628628130404500, 1.97529709676451637081125135565, 2.27446841186633416719373384707, 3.71887600825931860487631442624, 4.71088225378315451728474525386, 5.24608275465868938671918711089, 6.13069987557207208474693413836, 6.73862341612796237656813818385, 7.47425803148862222758839823334, 8.201869185986356142379883995436

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