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 + 3·7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 3·14-s − 15-s + 16-s + 17-s − 18-s − 20-s + 3·21-s − 22-s + 23-s − 24-s + 25-s + 27-s + 3·28-s − 9·29-s + 30-s + 5·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.223·20-s + 0.654·21-s − 0.213·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.566·28-s − 1.67·29-s + 0.182·30-s + 0.898·31-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

\(L(1)\) \(\approx\) \(1.985690828\)
\(L(\frac12)\) \(\approx\) \(1.985690828\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−17.45030444615089, −17.19748720445837, −16.23996590705912, −15.91049460014341, −14.91755672848924, −14.78060660699831, −14.17244318395239, −13.28464664208553, −12.69291406028564, −11.79697408237297, −11.40769635045110, −10.82823248726204, −10.02912177711376, −9.387137396750644, −8.688977089899347, −8.200010519200583, −7.556411339786480, −7.112720098947249, −6.091259323094525, −5.243307387936656, −4.353641672039494, −3.654551344098357, −2.618662662038429, −1.790816735317951, −0.8472598940732136, 0.8472598940732136, 1.790816735317951, 2.618662662038429, 3.654551344098357, 4.353641672039494, 5.243307387936656, 6.091259323094525, 7.112720098947249, 7.556411339786480, 8.200010519200583, 8.688977089899347, 9.387137396750644, 10.02912177711376, 10.82823248726204, 11.40769635045110, 11.79697408237297, 12.69291406028564, 13.28464664208553, 14.17244318395239, 14.78060660699831, 14.91755672848924, 15.91049460014341, 16.23996590705912, 17.19748720445837, 17.45030444615089

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