Properties

 Label 2-1800-1.1-c3-0-36 Degree $2$ Conductor $1800$ Sign $1$ Analytic cond. $106.203$ Root an. cond. $10.3055$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Origins

Dirichlet series

 L(s)  = 1 + 24·7-s + 28·11-s + 74·13-s + 82·17-s + 92·19-s + 8·23-s + 138·29-s + 80·31-s − 30·37-s − 282·41-s − 4·43-s + 240·47-s + 233·49-s − 130·53-s − 596·59-s − 218·61-s + 436·67-s − 856·71-s + 998·73-s + 672·77-s − 32·79-s − 1.50e3·83-s + 246·89-s + 1.77e3·91-s − 866·97-s − 270·101-s + 1.49e3·103-s + ⋯
 L(s)  = 1 + 1.29·7-s + 0.767·11-s + 1.57·13-s + 1.16·17-s + 1.11·19-s + 0.0725·23-s + 0.883·29-s + 0.463·31-s − 0.133·37-s − 1.07·41-s − 0.0141·43-s + 0.744·47-s + 0.679·49-s − 0.336·53-s − 1.31·59-s − 0.457·61-s + 0.795·67-s − 1.43·71-s + 1.60·73-s + 0.994·77-s − 0.0455·79-s − 1.99·83-s + 0.292·89-s + 2.04·91-s − 0.906·97-s − 0.266·101-s + 1.43·103-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1800$$    =    $$2^{3} \cdot 3^{2} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$106.203$$ Root analytic conductor: $$10.3055$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1800,\ (\ :3/2),\ 1)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$3.610178533$$ $$L(\frac12)$$ $$\approx$$ $$3.610178533$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
good7 $$1 - 24 T + p^{3} T^{2}$$
11 $$1 - 28 T + p^{3} T^{2}$$
13 $$1 - 74 T + p^{3} T^{2}$$
17 $$1 - 82 T + p^{3} T^{2}$$
19 $$1 - 92 T + p^{3} T^{2}$$
23 $$1 - 8 T + p^{3} T^{2}$$
29 $$1 - 138 T + p^{3} T^{2}$$
31 $$1 - 80 T + p^{3} T^{2}$$
37 $$1 + 30 T + p^{3} T^{2}$$
41 $$1 + 282 T + p^{3} T^{2}$$
43 $$1 + 4 T + p^{3} T^{2}$$
47 $$1 - 240 T + p^{3} T^{2}$$
53 $$1 + 130 T + p^{3} T^{2}$$
59 $$1 + 596 T + p^{3} T^{2}$$
61 $$1 + 218 T + p^{3} T^{2}$$
67 $$1 - 436 T + p^{3} T^{2}$$
71 $$1 + 856 T + p^{3} T^{2}$$
73 $$1 - 998 T + p^{3} T^{2}$$
79 $$1 + 32 T + p^{3} T^{2}$$
83 $$1 + 1508 T + p^{3} T^{2}$$
89 $$1 - 246 T + p^{3} T^{2}$$
97 $$1 + 866 T + p^{3} 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

−8.747140754500368337334338592119, −8.198097969563026750597942019560, −7.47420485954968219300071796335, −6.47425184975306120746819356963, −5.65661335678145225007903650797, −4.85762690659868484546026398669, −3.90514152228492132558566171605, −3.05847972843586962763972849065, −1.52132763296555907422716706388, −1.06578088634113477398216061899, 1.06578088634113477398216061899, 1.52132763296555907422716706388, 3.05847972843586962763972849065, 3.90514152228492132558566171605, 4.85762690659868484546026398669, 5.65661335678145225007903650797, 6.47425184975306120746819356963, 7.47420485954968219300071796335, 8.198097969563026750597942019560, 8.747140754500368337334338592119