Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7^{2}$ Sign $1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 0

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Dirichlet series

 L(s)  = 1 + 0.236·2-s − 3·3-s − 7.94·4-s + 5·5-s − 0.708·6-s − 3.76·8-s + 9·9-s + 1.18·10-s − 50.4·11-s + 23.8·12-s + 80.9·13-s − 15·15-s + 62.6·16-s − 76.3·17-s + 2.12·18-s − 4.13·19-s − 39.7·20-s − 11.9·22-s − 204.·23-s + 11.2·24-s + 25·25-s + 19.1·26-s − 27·27-s − 91.1·29-s − 3.54·30-s − 198.·31-s + 44.9·32-s + ⋯
 L(s)  = 1 + 0.0834·2-s − 0.577·3-s − 0.993·4-s + 0.447·5-s − 0.0481·6-s − 0.166·8-s + 0.333·9-s + 0.0373·10-s − 1.38·11-s + 0.573·12-s + 1.72·13-s − 0.258·15-s + 0.979·16-s − 1.08·17-s + 0.0278·18-s − 0.0499·19-s − 0.444·20-s − 0.115·22-s − 1.85·23-s + 0.0960·24-s + 0.200·25-s + 0.144·26-s − 0.192·27-s − 0.583·29-s − 0.0215·30-s − 1.14·31-s + 0.248·32-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$735$$    =    $$3 \cdot 5 \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : $\chi_{735} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 735,\ (\ :3/2),\ 1)$$ $$L(2)$$ $$\approx$$ $$1.014644887$$ $$L(\frac12)$$ $$\approx$$ $$1.014644887$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + 3T$$
5 $$1 - 5T$$
7 $$1$$
good2 $$1 - 0.236T + 8T^{2}$$
11 $$1 + 50.4T + 1.33e3T^{2}$$
13 $$1 - 80.9T + 2.19e3T^{2}$$
17 $$1 + 76.3T + 4.91e3T^{2}$$
19 $$1 + 4.13T + 6.85e3T^{2}$$
23 $$1 + 204.T + 1.21e4T^{2}$$
29 $$1 + 91.1T + 2.43e4T^{2}$$
31 $$1 + 198.T + 2.97e4T^{2}$$
37 $$1 - 155.T + 5.06e4T^{2}$$
41 $$1 - 156.T + 6.89e4T^{2}$$
43 $$1 - 354.T + 7.95e4T^{2}$$
47 $$1 - 175.T + 1.03e5T^{2}$$
53 $$1 - 200.T + 1.48e5T^{2}$$
59 $$1 - 312.T + 2.05e5T^{2}$$
61 $$1 - 154.T + 2.26e5T^{2}$$
67 $$1 - 734.T + 3.00e5T^{2}$$
71 $$1 + 678.T + 3.57e5T^{2}$$
73 $$1 - 60.8T + 3.89e5T^{2}$$
79 $$1 + 1.28e3T + 4.93e5T^{2}$$
83 $$1 + 116.T + 5.71e5T^{2}$$
89 $$1 - 916.T + 7.04e5T^{2}$$
97 $$1 - 1.41e3T + 9.12e5T^{2}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

Imaginary part of the first few zeros on the critical line

−10.09822262855160232908277404681, −9.101716728312629157979502679994, −8.393794864784479190534610305826, −7.46432206005037281795695406249, −5.98002292838930365049363290212, −5.72722030850673878027078542275, −4.52672211296869160804231260173, −3.70316437819187515230033112303, −2.10565151223534747762414395514, −0.57884463335329817099894324883, 0.57884463335329817099894324883, 2.10565151223534747762414395514, 3.70316437819187515230033112303, 4.52672211296869160804231260173, 5.72722030850673878027078542275, 5.98002292838930365049363290212, 7.46432206005037281795695406249, 8.393794864784479190534610305826, 9.101716728312629157979502679994, 10.09822262855160232908277404681