Properties

Degree $2$
Conductor $3630$
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 + 4·7-s + 8-s + 9-s − 10-s + 12-s − 2·13-s + 4·14-s − 15-s + 16-s − 6·17-s + 18-s + 4·19-s − 20-s + 4·21-s + 24-s + 25-s − 2·26-s + 27-s + 4·28-s + 6·29-s − 30-s + 8·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.554·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3630\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3630} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3630,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.191220356\)
\(L(\frac12)\) \(\approx\) \(4.191220356\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−18.20288667560097, −17.60966656961410, −17.24407915018714, −16.00606609708029, −15.77090195982003, −15.03690797876976, −14.48226004809722, −14.03256927933448, −13.43231212840224, −12.65484461427122, −11.87504216955601, −11.46797155823690, −10.81766651121423, −10.03499035797012, −9.089743725261433, −8.349411185370473, −7.820287229123506, −7.162445257086287, −6.350619868594546, −5.247587103857252, −4.563130974735256, −4.188823982603415, −2.921548308202113, −2.262918520553920, −1.132562207115698, 1.132562207115698, 2.262918520553920, 2.921548308202113, 4.188823982603415, 4.563130974735256, 5.247587103857252, 6.350619868594546, 7.162445257086287, 7.820287229123506, 8.349411185370473, 9.089743725261433, 10.03499035797012, 10.81766651121423, 11.46797155823690, 11.87504216955601, 12.65484461427122, 13.43231212840224, 14.03256927933448, 14.48226004809722, 15.03690797876976, 15.77090195982003, 16.00606609708029, 17.24407915018714, 17.60966656961410, 18.20288667560097

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