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: $\chi_{5610} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9199306675\)
\(L(\frac12)\) \(\approx\) \(0.9199306675\)
\(L(\frac{3}{2})\) 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

−17.50098771378258, −17.23051198317689, −16.49484543074848, −15.90828581148754, −15.21146553311659, −14.62055238211870, −14.07170431188789, −13.20700653755423, −12.51255855685391, −11.98170362734938, −11.36438847475834, −10.61405601553966, −10.22007343125952, −9.630840558283707, −8.722251352618682, −8.201869185986356, −7.474258031488622, −6.738623416127962, −6.130699875572072, −5.246082754658689, −4.710882253783155, −3.718876008259319, −2.274468411866334, −1.975297096764516, −0.5650664704928963, 0.5650664704928963, 1.975297096764516, 2.274468411866334, 3.718876008259319, 4.710882253783155, 5.246082754658689, 6.130699875572072, 6.738623416127962, 7.474258031488622, 8.201869185986356, 8.722251352618682, 9.630840558283707, 10.22007343125952, 10.61405601553966, 11.36438847475834, 11.98170362734938, 12.51255855685391, 13.20700653755423, 14.07170431188789, 14.62055238211870, 15.21146553311659, 15.90828581148754, 16.49484543074848, 17.23051198317689, 17.50098771378258

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